quest vs ask

quest is a noun but ask is not a noun.

Word	Noun	Adjective	Verb	Adverb
quest	Yes	No	Yes	No
ask	No	No	Yes	No

As verbs, ask is a hyponym of quest; that is, ask is a word with a more specific, narrower meaning than quest:

quest (noun)	ask (noun)
the act of searching for something
a search for an alternative that meets cognitive criteria

quest (verb)	ask (verb)
seek alms, as for religious purposes	make a request or demand for something to somebody
express the need or desire for	direct or put; seek an answer to
bark with prolonged noises, of dogs	address a question to and expect an answer from
search the trail of (game)	consider obligatory; request and expect
make a search (for)	require or ask for as a price or condition
	make a date
	require as useful, just, or proper