The phrase I would not call the Queen my aunt means I am happy with my situation (i.e., so much so that even becoming royalty could not improve on it).

For example, the following is from the Evening Herald (Dublin, County Dublin, Ireland) of 22^nd October 2003:

We asked six well-known young Dubliners to describe five things that make them happy. . .
[…]
► Alison Comyn,
former UK TV travel presenter, who is currently writing her first novel:
[…]
► My Five series BMW. Wooden trims, air conditioning, Lyric FM. When I sit in it, I wouldn’t call the Queen my aunt. I absolutely love it.

The earliest occurrence of the phrase that I have found is in the extended form I would not give sixpence to call the Queen my aunt, which, in its context, seems to mean I am not influenced by considerations of personal advantage. This occurrence is from the address that one C. G. Round delivered during the annual meeting of the Hinckford Agricultural and Conservative Club on 10^th November 1843—as transcribed in The Essex Standard, and General Advertiser for the Eastern Counties (Colchester, Essex, England) of 17^th November 1843 (C. G. Round specifies that the expression is “common”):

I for one earnestly trust that the future measures of the Government will be such that I shall be able to lend them my independent support. I, for my part, believe that they will be so; and I do believe that while the Government continue in its present course, lending themselves to no extremes, moderating between all parties, upholding the power of the country abroad, preserving peace at home, maintaining the national credit, and endeavouring to promote the vigour and prosperity of the country, they will receive the support of all moderate men. (Cheers.) The vast majority of the supporters of the present Government have no interest in being so; they have nothing to hope for— nothing in view from it. For my own part I can only say that I can look for nothing in return, and I would not, to use a common expression, give sixpence to call the Queen my aunt. (Laughter and cheers.) All that I desire is to obtain for my country the benefit of a good Government, to satisfy my own conscience, and, if possible, by all means to be approved of by such men as I have now the honour of addressing. (Cheers )

The second-earliest occurrence of the phrase that I have found means I would be content with my situation; it is from a letter that one Terry Driscoll wrote from Stoneybatter, near Dublin, to his friend, Mr. O Donohoe, St. Giles’s, London, on 12^th September 1850—letter published in The King’s County Chronicle (Parsonstown (modern-day Birr), County Offaly, Ireland) of 18^th September 1850:

I wouldn’t call the Queen my aunt this minnit, if I only had a rowl o’ notes on my groin, and could just tatther off to the Belfast, where there’s a power of divarshin’ goin’ on, in ordher to gratify the Lord Lieutenant; bonefires, and dancin’, and speechin, and eatin’ and dhrinkin’, of coorse (that’s a fayture people never omits in any sort of jubilee).

The English antiquarian and novelist Edward Peacock (1831-1915) recorded the phrase in A Glossary of Words used in the Wapentakes of Manley and Corringham, Lincolnshire (London: Published for the English Dialect Society, by Trübner & Co., 1877):

Queen, The, to call her my aunt. A phrase signifying the greatest honour or distinction that can happen to any one. An old woman at Winterton, who was receiving parish relief, said, ‘Oh, sir, if th’ board would nobut [= only] put me on another sixpence a week I wod n’t thank ye to hev th’ queen for my aunt.’

Note: The English Dialect Dictionary (Volume IV. M—Q – London: Published by Henry Frowde, 1903), edited by the English philologist and dialectologist Joseph Wright (1855-1930), recorded to call the Queen one’s aunt and to have the Queen for one’s aunt, which it defined as meaning to have the greatest happiness or distinction possible; it borrowed and slightly modified the quotation from Edward Peacock’s glossary:

An old woman at Winterton, who was receiving parish relief, said, ‘Oh, sir, if th’ boärd wo’d nobut put me on anuther sixpence a week I wo’d n’t thenk yě to hev th’ Queen for my aunt.’

The phrase occurs in this passage about “the regimental ladies of a lower grade” from Colonel de Crespigny’s Sweetheart, by John Strange Winter, pseudonym of the British novelist Henrietta Eliza Vaughan Stannard (1856-1911)—as published in The Illustrated London News (London, England) of 26^th May 1888:

“I wouldn’t,” remarked another of these ladies one day, after the Colonel had just discovered, or thought he discovered, a bad smell in her quarters, and, not being able to find it, had vented his disgust by asking her when she had washed her face last—“I wouldn’t be the Colonel’s servant, no, not to call the Queen my aunt. How that pore feller manages to keep his ’ead on his shoulders is more than I can tell. ’Pon my word, there’s never no pleasing of him. Here am I, all as clean as a new pin; even he can’t find no fault, except to fancy a smell; so he just finds fault with my complexion, and asks me when I’d washed my face last!”
“’E never did?” gasped one of her hearers.
“Oh! but didn’t he! And me as clean as a new pin!”
“And what did you say, Mrs. Morrisy?” asked a bystander, with breathless interest.
“Well,” returned Mrs. Morrisy; “I aint that afraid of the Colonel as some of the married ladies are; so I just up and says, ‘Some folk are born fresh-coloured, Sir, and some folk aint. I was born sallow.’ And then I gives a look at his great red face, and he gives a grunt and out he goes. Oh, I aint so scared of the Colonel as some I know; but, all the same, I wouldn’t be his servant—no; not to call the Queen my aunt!”

The phrase also occurs as the caption to this illustration for Colonel de Crespigny’s Sweetheart, published in The Illustrated London News of 26^th May 1888:

“I wouldn’t be the Colonel’s servant, no, not to call the Queen my aunt!”

The English author James Skipp Borlase (1839-1902) used a variant of the phrase with Queen Victoria in Poor Kitty M‘Knight: Or, Mines and Miners. A Tale of the Blantyre Colliery Explosion, published in The Hamilton Herald and Lanarkshire Weekly News (Hamilton, Lanarkshire, Scotland) of 13^th November 1903:

“Hurrah!” exclaimed Will Rugg, as his gaze rested with joy upon the numerous collieries, “this feels to me like coming home, after a long and weary wandering in strange countries. Once let me get a pick in my hand, and I wouldn’t give a fig to be able to call Queen Victoria my aunt.”