Quick or quickly on a question.

The second is grammatically more logical as quickly modifies the verb and so you need an adverb not an adjective. But in informal speech, as you say, the adjective is often used. I suspect it reflects a unconscious parallel with fast, which can of course be used as both adjective and adverb.

So - if you're speaking informally then either would be appropriate. In contexts where informality would be inappropriate (a job interview, say) and you need a more neutral style then I'd use quickly.