search vs quest after

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

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

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

search (noun)	quest after (noun)
the activity of looking thoroughly in order to find something or someone
boarding and inspecting a ship on the high seas
an investigation seeking answers
the examination of alternative hypotheses
an operation that determines whether one or more of a set of items has a specified property

search (verb)	quest after (verb)
inquire into	go in search of or hunt for
try to locate or discover, or try to establish the existence of
subject to a search
search or seek