Quiz

[snopes]

Comment: Is the following true?

A man was trying to convince an acquaintance that one individual can make
a lasting impression on others. After a rather heated discussion, his
friend continued to doubt this principle. To prove his point, the first
man declared that he would introduce a new word into the English language.
That night he chalked on walls and pavements throughout Dublin, Ireland,
four letters Q-U-I-Z, which he had chosen at random from the alphabet. The
next morning everyone who saw this unusual expression was baffled at it.
One person after another would ask, “What does it mean?” It wasn’t long
until the newspapers took up the question, and eventually this
strange-sounding word was on the lips of everyone. Thus the term “quiz”
was incorporated into the language as a synonym for “questioning.” The
originator had won his argument. He had left a lasting impression on many
people by creating one new English word.

[Logoboros]

According to the OED, the noun form of quiz shows up in the late 18th century to mean a eccentric person or a prankster, and by extension, a prank or joke. This may well come from the verb quiz (dating from about the same time), meaning to mock or tease. It's also used to mean "a) To regard with amusement or scorn; to appraise mockingly; (b) to peer inquisitively at" -- in which sense, it may derive from inquisitive.

The specific sense of "a set of questions," the OED describes as "originally and chiefly North American."

However, the OED does reference the anecdote in the OP:

The source of quot. 1793 at sense 1b suggests that the word was regarded as unusual at an early date. An origin in public school slang may be suggested by the following:

1798 G. Colman Heir at Law iv. iii. 60 A gig? Umph! that's an Eton phrase—the Westminster call it Quiz.

The following anecdote is widely repeated, but cannot be confirmed. It apparently first appears in Smart (1836), but is omitted in the 1840 edition. The man in question is perhaps Richard Daly (1758–1813), Irish actor and (from 1780) theatre manager:

1836 B. H. Smart Walker Remodelled at Quiz, Daly, the manager of a Dublin playhouse, wagered that a word of no meaning should be the common talk and puzzle of the city in twenty-four hours; in the course of this time the letters Q, u, i, z were chalked or pasted on all the walls of Dublin with an effect that won the wager.

ETA:
The OED also notes that quiz might come from quoz (a word that appears at the same time and with the same meaning of "an odd or ridiculous person or thing"). Under the representative quotes for quoz we find these two interesting examples:

1796 F. Burney Camilla IV. vii. xiii. 200 ‘The quoz of the present season are beyond what a man could have hoped to see!’ ‘Quoz! What's Quoz, nephew?’‥‘Sometimes we say quiz, my good sir.’
1802 in Spirit of Public Jrnls. (1803) 6 197 At length it was announced, that Pic-Nic, like Quoz, which was chalked some years ago on windows and doors, really meant nothing.

The first quote might suggest that there was some emerging slang word, which perhaps if the Daly anecdote is true, perhaps pre-existed his "q-u-i-z," but which gets conflated with an amplifies (or is amplified by) his Dublin prank. The second quote actually does suggest that something akin to the anecdote does lie in the word's origins (it's amusing that it's mentioned in the same breath as the similarly etymologically fraught picnic), and it strikes me as a bit odd that the discussion in the OED's etymology for quiz doesn't mention this as potential support for the anecdote.

At any rate, the "synonym for 'questioning'" angle in the OP seems far-fetched. But referring to a string of chalked up letters as a prank, is rather more plausible.

[Embra]

If the prankster did choose his four letters at random, he was lucky to get a U to go with his Q...

[Dr. Dave]

That was my exact thought, as well as the good fortune of getting one other letter that acts as a vowel, and that letter appearing between consonants.

-"Math Qzui" Dave