seek vs quest after

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

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

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

seek (noun)	quest after (noun)
the movement of a read/write head to a specific data track on a disk

seek (verb)	quest after (verb)
inquire for	go in search of or hunt for
try to locate or discover, or try to establish the existence of
go to or towards
try to get or reach
make an effort or attempt