The phrase Queen Anne’s fan designates a gesture of derision made by putting one’s thumb to one’s nose and outspreading the fingers like a fan; this gesture can be intensified by joining the tip of the little finger to the thumb of the other hand, whose fingers are also outspread fanwise.
—Cf. also to cock a snook.

This phrase probably refers to Anne (1665-1714), Queen of England and Scotland (known as Great Britain from 1707) and Ireland, who reigned from 1702 to 1714. However, the motivation for the choice of her name is unclear.
—Cf. also Queen Anne front and Mary Ann back.

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

1-: From the Aberdeen Evening Express (Aberdeen, Aberdeenshire, Scotland) of Friday 26^th January 1883:

What is known by bad boys as Queen Anne’s fan—a vulgar attitude, in which the fingers and the nose are prominent.

2-: From Wellington Petty Sessions, published in The Wellington Journal and Shrewsbury News (Wellington and Shrewsbury, Shropshire, England) of Saturday 12^th March 1887:

Alleged Threats.—John Robinson, innkeeper, Coalpit Bank, was summoned for having used threats towards Sarah Wainwright.—Mr. Carrane represented the complainant, and Mr. Littlewood the defendant.— Complainant said that on the day in question she was standing outside her door at Hadley talking to a neighbour, and the defendant rode by on a horse. As soon as he got up to where she was he put his thumb to his nose and made a “Queen Anne’s fan,” and she remarked to her friend, “That is the man who took our horse.” He used very bad language, and said he would ride over the top of her and kill her.

3-: From The Weekly Irish Times (Dublin, County Dublin, Ireland) of Saturday 15^th June 1901:

ABREAST OR TANDEM.

The late Archbishop Magee once had a complaint made to him that a certain clergyman in his diocese was in the habit of driving tandem, to the horror of his parishioners. The Bishop (as he then was) was asked to remonstrate with him. He accordingly did so.
“It is ridiculous,” said the clergyman. “If I drove a pair, nothing would be said, but because I simply put one horse before the other I am hauled over the coals. I don’t see the difference myself.”
“Look here,” said the Bishop, solemnly, “if I were going to say my prayers, would you say it was all this same thing if I put my hands together, so or (gravely making the well known Queen Anne’s fan) placed them one before the other—so?”

4-: From The Mid-Sussex Times (Haywards Heath, Sussex, England) of Tuesday 10^th September 1907:

QUEEN ANNE’S FAN.

At Eastbourne Mr. Oscar Lewisohn, of Forwood, Ascot, husband of “Miss Edna May,” was summoned for driving a motor-car at a dangerous pace and with failing to stop when signalled to by a police-constable. The defendant set a plea of “Not guilty.” Constable Ashenden said he saw a car coming from the railway station towards the sea at the rate of about 20 miles an hour. He stepped into the road and held up his right hand to stop the car. The defendant, who was driving, placed his thumb to his nose in an insulting manner, slackened speed, but then drove on. Mr. Newton, in defence, said his client was a gentleman of position, and would be the last person in the world to avoid proper authority. Mr. Lewisohn’s driver swore that his master did not put his fingers to his nose. The speed, he said, was only 15 miles an hour. Two previous convictions were proved, and the defendant was fined £25, including costs.

5-: From Fun, Facts & Fancies, published in The Isle of Wight Observer (Ryde, Isle of Wight, England) of Saturday 29^th May 1915:

In the slang of the time, “Queen Anne’s fan,” was what the rude youngsters of our own days call taking a sight, i.e., with the thumb to the nose and the fingers spread. Why so termed we have no idea.

6-: From the Lincolnshire Echo (Lincoln, Lincolnshire, England) of Monday 21^st June 1915:

“Queen Anne’s fan” was the polite name for the action which consists of putting one’s thumb to the nose and wiggling the four fingers in derision.

7-: From John Bull (London, England) of Saturday 11^th September 1915:

Landlord and Tenant.
A benevolent Tottenham landlord had been worried in his mind ever since last December, because he hated to think of making his tenants pay for the increased income tax on his property. But there was little use in postponing the evil day; the dreadful thing had to be done at last, so Mr. Tenant was informed by letter that his rent was “up” one shilling a week, as he, Mr. Landlord, was “obliged to recoup himself for this extra outlay.” And he said, “I wish you to note that I shall not benefit anything from this, but shall have to pay it away in extra income tax and local rates.” If he gets a shilling a week, at the least, from each tenant of his, he will probably be making more money than he pays away in fresh taxation. But the amusing point in this piece of extortion is the calm way that a landlord assumes that somebody else must pay his dues for him. Extra income tax is imposed by Government on the person owing the income; the purpose of the State is immediately nullified by the liability being passed on the poor people who are already paying their own taxes. In the present case, the notice happens to be addressed to a tenant who holds an agreement that does not expire till 1916, and who, instead of paying the extra shilling, may apply his fingers to his nose in the outspread fashion known as a Queen Anne’s fan.

8-: From A dictionary of English phrases: phraseological allusions, catchwords, stereotyped modes of speech and metaphors, nicknames, sobriquets, derivations from personal names, etc.: with explanations and thousands of exact references to their sources or early usage (London: George Routledge & Sons, Ltd., 1922), by the British civil servant and historian Albert Montefiore Hyamson (1875-1954):

Queen Anne’s fan: an offensive gesture, made by the thumb and fingers spread out from the nose.