No, you need an adverb; it would have to be 'quickly', Taka.
Isn't it possible to use "quick" adverbially, as in, say, "Do it quick!"?
In that particular context (i.e., Do it quick!), it's possible, iff quick refers to 'quick-like'. In that case, 'quick' would be an adjective in form but an adverb in function, a near synonym with "right now, this very instant", like this,
EX: Do it quick(-like)! (i.e., This very instant, now!)
Do it when/In what manner?
Quick-like (i.e., right now, this very instant).
As for the context in question, though, MM is correct in pointing out that "he speaks _____" requires an adverb,
He speaks quick. When/In what manner does he speak? He speaks quick-like. (i.e., right now, this very instant) (Not OK)
Note, colloq. He speaks *quick ~ He speaks too fast.
He speaks quickly, very rapidly. (adverbs) In what manner does he speak?