This is the definition of the obsolete Australian-English phrase Fiji uncle, uncle (who lives) in Fiji, etc., from A Dictionary of Australian Colloquialisms (Sydney University Press in association with Oxford University Press Australia, 1990), by Gerald Alfred Wilkes (1927-2020):

An imaginary rich uncle overseas, backing some venture in which the unwary may be persuaded to invest.

(The Fiji Islands, in the South Pacific, consist of some 840 islands. First visited by Abel Tasman in 1643, they became a British Crown Colony in 1874.)

The phrase had become proverbial by the early 20^th century—as shown by “a little friendly advice to the innocent traveller” in versified form published in The Express and Telegraph (Adelaide, South Australia) of Wednesday 2^nd April 1902:

Oh the uncle who lives in Fiji
Is a mythical sort of a chap,
And you take it as gospel from me
That his nephew is not worth a rap.
For his legacy’s still in the air,
And no trace of it e’er will he see,
So again let me bid you beware
Of the uncle who lives in Fiji.

And that man is a palpable fraud,
When he offers you part of his wealth
For a grab at your own little hoard,
On the ground he’s good-natured by stealth.
For his pockets are probably void,
Tho’ his wealth he continues to laud,
It’s a fiction he’s often employed
For the man is a palpable fraud.

And beware of the vanishing friend,
He will leave you at last in the lurch;
He is itching your money to spend,
Tho’ you think him as “safe as a church.”
For he’s off like a shot from a gun,
And you’ll find out his game in the end;
Like a dinner you’ll surely be done,
So beware of the vanishing friend.

These are the earliest occurrences of the phrase that I have found, in chronological order:

1-: From The Australian Star (Sydney, New South Wales) of Wednesday 9^th May 1888:

A GUILELESS QUEENSLANDER.
Tricked out of £318.

After several remands Henry Ford, a respectably dressed man, was charged at the Central yesterday with obtaining by trick the sum of £318 from James Reed. Senior Constable Power arrested Ford on April 23. James Reed said he was an ex-publican, but at present was residing in Charlotte-place. On April 20 he met prisoner in George-street. Prisoner knocked up against him and begged his pardon. The apology led to conversation, conversation led to a walk, the walk led to the Mint, which Reed said he would like to see, as he was a stranger to Sydney. Ford said he had tickets, but they did not go through the building, and subsequently they found themselves in a public house near Circular Quay, where another strange gent joined them. During the conversation that ensued Reed appears to have confided the knowledge that he hailed from Croydon to his new-found friends. The newcomer was a tall, thin individual, with a dark moustache. He appeared to enjoy the joke immensely, for he laughed immoderately, said he was in a circus, and only six or eight months in the country; that his uncle in Fiji sent him £500 to take him and his brother out. Also he mentioned that his uncle had died, and left him and his brother an estate and £100,000. For fear any doubt would rest in the simple mind of Mr. Reed, the heir to the property mentioned that at that present moment he had £700 in his pocket. In reference to the conversation about the circus the stranger related an account of a spree he indulged in the previous night, and how he lost £32 by a trick in which a sea captain won a little game that was played by the aid of eight shells. The stranger made out that he had been had, and prisoner appeared to sympathise with him and declared that the stranger had been let in. The latter denied it, and the prisoner asked the stranger to give him and Reed the same chance and see how it was done. The prisoner then got eight pieces of paper and put them on the table. He then took a sovereign from his pocket and put it in the stranger’s hat on the table. The stranger pulled a roll of notes from his pocket and put one in the hat with the sovereign. Prisoner asked witness if he would go in too, but he declined. “Very well,” said prisoner, “you be referee.” Witness replied, “all right.” Prisoner won the game. The game was played with eight pieces of paper and two coins, one coin being hidden on one side by prisoner and the prosecutor, who turned up head or tail, and the other being tossed up by the stranger in his hand. The stranger said, “Whatever this comes, I’ll call.” The stranger’s call always corresponded with the coin the others had turned up, and whoever won took away one piece of paper from his own number. The stranger had five pieces, and the prisoner three. Prisoner won the game, and took the money out of the hat. He showed witness the note, which appeared to be a £50. Witness advised prisoner to return the £50. The stranger pretended to be under the impression that the note was a £1 until witness told him. They then persuaded witness to put £10 in the game. The others put in £10 each. During the game the stranger said he thought the others had no more cash, so withdrew his tenner. This fetched witness, who declared he could show £200 or more to-morrow. The stranger agreed to give them £50 each if they could produce the sum next day at Redfern railway station. Prisoner took the £50 from the stranger to take care of. They met next day, but Reid [sic] could not get his money from the bank, and the following day he drew £318 from the bank, and met the prisoner and the stranger at a hotel at the railway station. The paper trick was played again until £1840 was in the hat, £318 of which belonged to witness, the rest being cheques. The stranger won the £318 and the remaining cheques, and shouted, and cleared out. Prisoner afterwards gave witness £5, and they parted, after promising to meet again and go to Queensland. Prisoner said they had been forging cheques. Subsequently, prisoner gave witness £3 at the railway station. There they parted to meet in court.
Prisoner was committed to the Quarter Sessions for trial; bail, self in £300, with two sureties in £150 each, or one in £300.

2-: From the Evening News (Sydney, New South Wales) of Wednesday 24^th November 1897:

The Old, Old Story.
A GUILELESS COUNTRYMAN
LOSES £500.

In February last a young farmer from near Coonamble came to Sydney with a view of doing business with the wool merchants. Whilst looking about the city he was met by a man who said he was a squatter from St. George. They talked about wool, and became very friendly, and the new acquaintance proposed to show the stranger round the city. They met a third party, and the trio went round the usual city resorts. Finally they found themselves at a hotel in Newtown, and while there an affable stranger walked in. He begged their pardon, and hoped he was not intruding. He said he had met a captain who showed him a game with five matches. He then showed them the game, and when they won £1 from him he threw them £5. They pointed out the mistake, and the “affable stranger” was greatly pleased with their apparent honesty. It was the old story about the rich Fiji uncle who directed him to assist any deserving man he might meet. He then asserted that for every £1 they could show he would give them another. One of the men said he had £1000. The other said that he had £300. They made an appointment for a week later on, and the young farmer met them in Victoria Park with his £500. After some preliminaries, they again indulged in the “match” game, and in about five minutes all the money was won by the “Fiji benefactor.” One of the men took pity on the young man, and lent him £50 of his own money, and promised to lend the balance when he got home. The simple-minded farmer appears to have done nothing until, within the last ten days, from something he heard, he came to Sydney and got a warrant out for the three alleged conspirators. Yesterday Detectives Roche and M’Lean arrested a man in connection with the affair. He was brought up on a charge of vagrancy, and remanded till to-day. He was identified by the prosecutor as one of the three men in question.

3-: From The Bendigo Independent (Bendigo, Victoria) of Tuesday 6^th September 1898:

THE ANTI-USURERS’ BILL.

People have short memories, especially those people who have no direct or immediate personal interest in the matter at issue. Those, however, who have interests to serve and irons to be heated at other people’s fires have very strong and active memories. It is thus with the Anti-Usurers Bill. Except by a few the evidence seems to have been forgotten that was taken by a select committee of the Legislative Council regarding the working of the usury system in Victoria. Most shameful extortion, equal to almost anything that was brought to light by the still more recent Commission on Law, was laid bare before the Usury Commission. The trickery and the viciousness of the system was such that on the report of the Commission being laid on the table of the Legislative Assembly it constrained such an aged and usually unemotional politician as the Hon. James Service to exclaim of a certain usurer that “He was a devil incarnate, a very prince of devils.” And as the Herald, which for many years has done such good, fearless and sensible work in the exposure of frauds and cheats of various descriptions has from time to time shown us, Mr. Service’s “Prince of Devils” had many other “devils” (large and small) to keep him in countenance, to see that the Satanic work does not flag and is not neglected now that the chief is no longer able to personally supervise it in Melbourne. There are still a few philanthropic “ladies” advertising in the Melbourne press that they are anxious to lend money to the impecunious. Several “retired gentlemen” are also aiding in the good work of the other “ministering angels.” What these and other individuals like them are discovered to be when followed up the Herald first and the Usury Commission subsequently, enlightened us. The philanthropic “ladies” and the “retired gentlemen” are so many traps and snares for plundering the unfortunate, the simple-minded or the reckless and foolish. Sometimes it has been shown that the alleged lender has no money to lend, or at least money lending is not in his line. A dupe caught by the advertisement of such an individual will state his case. The usury sharp will speak him fair words. He will state that if what the would-be borrower says is true, he does not see any objections in the way of making the advance required. But as it is not his own money that he is dealing with but the money of the widow lady, the retired gentleman, or the rich uncle from Fiji, enquiries must be made. The “enquiry fee” is 10s. or 20s. The dupe pays it to the “confidence man,” and it is the last he sees of it. On his next visit to the “confidence man’s” office he is told in effect that the “enquiries” have been made and that the “confidence man” does not feel justified in risking on such security any of the money of his clients—especially that of the rich uncle from Fiji. Of course a man cannot be termed a usurer who like this class of harpy does not lend anything whatever, but keeps himself afloat by “enquiry fees” and “travelling expenses” to inspect country properties which he never sees. But it throws light on the tricks of the trade, and it shows the extent to which this class of city sharp will go in order to avoid doing a day’s honest work with his hands or brains.
Though a few months ago the newspapers were full of the evidence that was being taken by the Usury Commission, and the bill which the Attorney-General now has before Parliament is based on the recommendations of that commission, a few interested parties have put their heads together and if the press and public do not speak out, and if members of Parliament do not do what they ought there is a risk of the measure being shelved. Mr. Isaacs is not being supported with this measure as he should be. There are deep and cunning wire-pullers behind the scenes, and none know this better than the members of Parliament themselves. Knowing the facts as they do they ought to be ashamed of themselves if they are drawn away from their duty to the public by the button-holing and the greedy importunities of the agents of the “widow lady,” the “retired gentleman” and the Fijian uncle.