Air quoting doesn’t have an equivalent in French, or at least didn’t have one until the double quotes tended to replace the standard French guillemets in casual writing (due to the French keyboard layout limitations) and make the fingers mimic more easily achievable.

However, even without mimicking the quotes, we can still say entre guillemets to express the idea that there is something special about what was just said.

It has already been discussed here.

“Air quotes” doesn’t have equivalent to french, because it’s in my mind a written transcription of “whats up today ?”. Explanation in the meaning of what is the new tendency … visiting Tokyo is one of the most “quoted” place to go … and on “Air” is concerning what’s today (august 2018) best place on earth to go in looking after what people may say about (talks spread thru atmosphere and definitely wrote on Air, if there are tagged or quoted high) ?
Pejoratively, if you can quote in the air of Tokyo something, the best way is to visit only … (lol). The two fingers up drawing “entre guillemets” like double quotes just mean in that case : For visiting … yes go, but it’s not the best place to live

“how do French speakers express this idea?”

One French speaker expressed it thus:

l’isoler dans une intonation speciale, machinale et ironique comme
si’il l’avait mise entre guillemets

Wordy, but, after all, it’s Proust.

A little further down the page, he’s more succinct:

mettre entre guillemets

It’s from Du Cote de Chez Swann page 98 in the old, 3-volume Pleiade edition of the Recherche

In context:

[Swann:] je ne crois pas beaucoup a la “hierarchie“

[Marcel:]
quand il parlait de choses serieuses, quand il employait une
expression qui semblait impliquer une opinion sur un sujet important,
il avait soin de l’isoler dans une intonation speciale, machinale et
ironique, come s’il l’avait mise entre guillemets, semblant ne pas
vouloir la prendre a son compte et dire “la hierarchie vous savez,
come disent les gens ridicules”