The obsolete noun quoz denoted an odd or ridiculous person or thing. It is first recorded in The Festival of Momus, a Collection of Comic Songs, Including the Modern and a Variety of Originals (London, 1780?).

The 1789 text entitled Quoz, a New Song (quoted below) seems to indicate that quoz was merely a fanciful variant of the noun quiz. Attested in 1780, quiz originally denoted a person whose appearance is peculiar or ridiculous—cf. origin of ‘quiz’ (“Vir bonus est quis?”)?.

In the following passage from Volume IV of Camilla: Or, A Picture of Youth (London, 1796), by the English author Frances Burney (1752-1840), Clermont Lynmere, an obnoxious student, uses both quoz and quiz when talking to Sir Hugh Tyrold about Mr. Westwyn, a good-hearted old man and a friend of Sir Hugh’s:

“Upon my honour,” cried Lynmere, piqued; “the quoz of the present season are beyond what a man could have hoped to see!”
“Quoz! what’s quoz, nephew?”
“Why, it’s a thing there’s no explaining to you sort of gentlemen; and sometimes we say quiz, my good old sir.”

The following song was published in the poetry section of The New London Magazine; Being an Universal and Complete Monthly Repository of Knowledge, Instruction, and Entertainment (London) of September 1789. This song inventories the fashionable words and phrases that quoz has recently supplanted—among them is quiz (cf. “Jack’s a Quiz; because Jack gives way to Jill, and so does Quiz to Quoz.”):

QUOZ,
A NEW SONG.
Written and sung by Mr. Edwin, at the Theatre-Royal, Haymarket.

HEY for such buckish words, for phrases we’ve a passion
Immensely great, and little once, were all the fashion;
Hum’d, and then humbug’d, Twaddle tippy poz;
All have had their day—but now must yield to Quoz.

Walk about the town, each time you turn your head, Sir,
Pop, staring in your phiz, is Q, U, O, and Z, Sir:
Cried Madam Dip to deary, its monstrous scandaloz,
To write on people’s shutters that shameful, nasty, Quoz.

Once it was the Barber, for ev’ry thing that’s right:
The Shaver knock’d the Barber quickly out of sight.
Now we’ve got a new word, how invented ’twas,
If you ask, I’ll tell—, my answer, Sir, is Quoz.

The Hobby Horse, of late, we rode about with speed,
For drinking, wenching, gaming, ’twas the word, indeed;
Then Macaroni, Bore, and Rage, never sure the like was,
Yet all that sort of thing gave way to little cunning Quoz.

Tipsy, dizzy, muzzy, sucky, groggy, muddled,
Bosky, blind as Chloe; mops and brooms, and fuddled,
Florid, torrid, horrid; stayboz, hayboz, layboz—
Words with terminations not so good as Quoz.

But when Quozzy came, Tippy, Bore, and Twaddle,
Bucks of blust’ring fame, could not keep their saddle:
One attempts to rally—bully Quiz it was;
But by nightly Sally, dubs him little Quoz.

There’s a jack to roast your meat, a jack to hold your liquor;
Jack upon the green to amuse the Vicar:
Jacks of various sorts—Jack’s a Quiz; because
Jack gives way to Jill, and so does Quiz to Quoz.

Some may think it French, some may call it Latin;
Some give in this meaning, others will give that in:
Mean it what it will, or sense or non compos,
The meaning, I should think—the meaning must be—Quoz.

Suppose we say its drinking—suppose it means a dinner—
Suppose a Methodist—suppose a wicked sinner—
To finish my suppose—suppose I make a pause—
I’ve hit it now—’tis thank ye—and so, good people, Quoz.

In the Preface to A Classical Dictionary of the Vulgar Tongue (London, 1785), the English antiquary and lexicographer Francis Grose (1731-91) wrote about the ephemeral nature of the fashionable words, and explained that he had recorded them—the words in bold appear in Quoz, a New Song:

The fashionable words, or favourite expressions of the day, also find their way into our political and theatrical compositions; these, as they generally originate from some trifling event, or temporary circumstance, on falling into disuse, or being superseded by new ones, vanish without leaving a trace behind, such were the late fashionable words, a Bore and a Twaddle, among the great vulgar, Maccaroni [sic] and the Barber, among the small; these too are here carefully registered.

In the same edition of A Classical Dictionary of the Vulgar Tongue, Francis Grose defined several of the words mentioned in Quoz, a New Song:

THE BARBER, or that’s the barber, a ridiculous and unmeaning phrase, in the mouths of the common people about the year 1760, signifying their approbation of any action, measure, or thing.
[…]
BORE, a bore, a tedious troublesome man or woman, one who bores the ears of his hearers with an uninteresting tale, a term much in fashion about the years 1780, and 1781.
[…]
MACCARONI, an Italian paste made of flour and eggs; also a fop, which name arose from a club, called the maccaroni club, instituted by some of the most dressey travelled gentlemen about town, who led the fashions, whence a man foppishly dressed, was supposed a member of that contraction stiled a maccaroni.

In Additions and Corrections, Grose defined another word mentioned in Quoz, a New Song:

Twaddle, perplexity, a confusion, or any thing else, a fashionable term that succeeded a bore.

Quoz, a New Song, was followed by this fanciful etymology of quoz in The Bury and Norwich Post; Or, Suffolk, Norfolk, Essex, and Cambridge Advertiser (Bury St Edmunds, Suffolk) of Wednesday 23^rd September 1789:

Origin of the present fashionable Word QUOZ.

Shortly after the destruction of the Bastile [sic], the most valorous men of France fled from their country, like so many lions, from the crowing of one solitary cock. Arriving in the Downs, on board some of the Dieppe fishing boats, they made signals for the Dover pilots to come off. When these people (who are justly stiled sharks) came on board the French vessels, they saw, by appearances, that the passengers were none of the common sort of men, they asked very exorbitant prices for bringing them and their baggage on shore; upon which the Frenchmen shrugged their shoulders, and fore and aft sent the general cry of “Quoi, quoi, quoi,” (in English, “What, what, what.”) The pilots immediately cried out, “Damn your Quoz, Quoz, Quoz, speak that we may understand you, and don’t bore us with your Parley vous and Quoz.”