“and I'm the Queen of Sheba”

Solution 1:

At the turn of the previous century, the Queen of Sheba was a crowd-pulling spectacular traveling show, the most famous actress was a Russian "Queen of Sheba". Solomon and the Queen of Sheba was one of the many themes such as Cleopatra included in travelling shows. By the 1910s the Ringling Bros. Circus had more than 1,000 employees, 335 horses, 26 elephants, 16 camels and other assorted animals that travelled on 92 railcars. "The Greatest Shows on Earth" included pageantry and other performances in addition to the ring.

These and similar spectaculars would have been very well known to G. B. Shaw when he wrote his plays that retold historical events in modern settings. One of Shaw's most successful plays, Pygmalion, written in 1912 was staged in Vienna the following year and in Berlin shortly afterwards. By 1914 the author of Pygmalion has become the most popular writer in England.

The story was later presented to the masses first as Pygmalion on the screen

and later became M(a)yFair Lady.

ELIZA [curiously] 'Ere, what's that you say?

HIGGINS. Yes, you squashed cabbage leaf. You disgrace to the noble architecture of these columns! You incarnate insult to the English language! I could pass you off as, er, the Queen of Sheba.

ELIZA [laughing] Ah-how-ow!

https://en.wikipedia.org/wiki/Pygmalion_(play)#Different_versions

Around the early 1930's there were several movie adaptations and in movie magazines there were Woolworths Stores advertisements for Embassy Powder,

"A $1 quality face powder for 20c? If that's true, I'm the Queen of Sheba."

Solution 2:

The first use of the expression

I am the Queen of Sheba,

by somebody who was not actually the queen of Sheba herself, that I can find on Google books is in the 1877 book

The Queen of Sheba, by Thomas Bailey Aldritch,

where it is used repeatedly by somebody who does not actually seem to be the Queen of Sheba. It doesn't appear to be used in the sense the expression is used in today—if that's true, then I'm the queen of Sheba. From skimming the book, it appears that she is mentally ill (at least, she has been put in an insane asylum).

So this probably explains why we generally use Sheba as the country in this snow clone, unless the expression was around before this book was written and the book is playing off it. I can't find any instances of this snow clone (either with the Queen of Sheba or with other people) in Google books before 1877, but I easily could have missed them.

Solution 3:

As a point of interest, the reverse case,

or I am the Queen of Sheba

wherein the contrary-to-fact claim serves to humorously assert the truth (rather than the falsehood) of the preceding statement, appears at least as early as 1833, in The Stolen Child, by Scottish novelist John Galt:

However, give yourself no uneasiness, you are my lord's brother or I am the Queen of Sheba.

The speaker, a Mrs Servit, is characterized by her use of Scottish idioms in conversation:

...her accent, however, betrayed her Scottish origin, and yet she did not exactly speak the language of the country, though with a few deviations she evinced a perfect mistressy of its most recondite and common idioms.

These circumstances, along with the nationality of the the author, although they do not establish the point, suggest that the contrary-to-fact claim "and/or I am the Queen of Sheba" may have been a Scottish idiom in the early 1800s.

The earliest similar use of "I am the Queen of England" uncovered by my research appears in the 1849 romance Trials of Love, by Mrs H.M. Lowndes née Hannah Maria Jones:

He's about as much of a footman as I am the Queen of England....

For "I am the King of Siam", my research turned up no uses in the 1800s or earlier.