Can you say "Quid of the situation?" to ask "What about the situation?"

Solution 1:

It's not something I've heard myself (as a native speaker of American English), but I found examples of "Quid of the x?" online. And all the examples I found have one thing in common: the author speaks French, or (when it's not clear if they speak French) at least lives in a country where French is popularly spoken.

And that's not insignificant. Quid is used en Français.

Examples:

Anybody with news about the new #HorizonEU launching date, submission procedures and also quid of the #Swiss participation? — https://mobile.twitter.com/david_billard/status/1400425641342869504?s=20

Note the French-style spacing before the question mark here:

Quid of the anti-tutsi propaganda and violence in this period ? — https://mobile.twitter.com/CahayJF/status/1301963280915877891?s=20

https://mobile.twitter.com/DameAilys/status/1380070143510913025?s=20

https://mobile.twitter.com/xrolet/status/1244228448240971780?s=20

https://mobile.twitter.com/HagueCA_EU/status/1248550689015574528?s=20

https://mobile.twitter.com/grainburger1/status/1270430635585404931?s=20

Note: "quid of the matter" is not used as a question.

Solution 2:

My Latin master at school would mix English and Latin words in a sentence for comic effect (along similar lines to Franglais) and as quid is Latin for what, this could be a vague memory of macaronic writings in the same vein.

Solution 3:

quid, quid of the matter

quid

That which a thing is. Cf. quiddity n.

quiddity Chiefly Philosophy. The inherent nature or essence of a person or thing; what makes a thing what it is.

The quid of the matter is found..in the fact that in the table..these authors have chosen to use three ‘identifiable’ forces. (OED)

quid (plural quids)

The inherent nature of something. Wiktionary

quiddity Whatever makes something the type that it is : ESSENCE

When it comes to synonyms of "quiddity," the Q's have it. Consider "quintessence," a synonym of the "essence of a thing" sense of "quiddity" (this oldest sense of "quiddity" dates from the 14th century). "Quibble" is a synonym of the "trifling point" sense; that meaning of "quiddity" arose from the subtler points of 16th-century academic arguments. And "quirk," like "quiddity," can refer to a person's eccentricities. Of course, "quiddity" also derives from a "Q" word, the Latin pronoun quis, which is one of two Latin words for "who" (the other is "qui"). "Quid," the neuter form of "quis," gave rise to the Medieval Latin quidditas, which means "essence," a term that was essential to the development of the English "quiddity." m-w

Surely the quid of the question lies in the willingness to constantly improve the quality of second language teaching and learning or at least to attempt to do so whenever possible because... Juan de Dios Martinez Agudo; Teaching and Learning English trhough Bilingual Education

Quid of the matter has a curious ngram with a notable peak in the 1915-1920 range. Quid of the situation does not appear.

And so a rupture developed with the group of women who had a program. That was the quid of the matter. Elena Pedraza, quoted in K. A. Rosemblatt; Gendered Compromises

The quid of the matter lies in the question of whether a colonial territory can be self-determining under the conditions that previal [sic] when all the colonial power's mechanisms of control... Tricontinental Magazine, Issues 51-57 p.37 (1977)

It was truly strange, because my imagination and obsessions never, ever bore me; on the contrary I bear them in mind for a long, long time. But I no longer could. No, and that was the matter, the quid of the matter, the heart of my disinterest . . . My imagination flitted about like a kite, without my realizing that I did not want to continue telling the story I already knew. Carmen Boullosa; The Perfect Novel

From his point of view, legitimate hierarchical relations can exist which are not domination relations. The quid of the matter lies in the legitimacy of power, and a government by discussion can be legitimate as long as it maintains a strengthening of the most vulnerable sectors as a priority task... J. A. Bidaguren; The Political Dimension of Local Human Development pp.33-34

This meaning of quid as essence differs from its use in referring to the "quid" part of a quid pro quo:

That was the quid of the matter. The quo was giving up the tenancies, and because of the position in that connection we wish to retain the 1976 Act. The Parliamentary Debates (1984)