quest vs go after

quest is a noun but go after is not a noun.

Word	Noun	Adjective	Verb	Adverb
quest	Yes	No	Yes	No
go after	No	No	Yes	No

As verbs, go after is a hypernym of quest; that is, go after is a word with a broader meaning than quest:

Other hypernyms of quest include chase, chase after, dog, give chase, tag, tail, track, trail.

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

quest (verb)	go after (verb)
seek alms, as for religious purposes	go in search of or hunt for
express the need or desire for	go after with the intent to catch
bark with prolonged noises, of dogs
search the trail of (game)
make a search (for)