lay claim vs quest

lay claim is not a noun while quest is a noun.

Word	Noun	Adjective	Verb	Adverb
lay claim	No	No	Yes	No
quest	Yes	No	Yes	No

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

lay claim: demand as being one's due or property; assert one's right or title to
quest: express the need or desire for

Other hypernyms of lay claim include ask for, bespeak, call for, request.

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

lay claim (verb)	quest (verb)
demand as being one's due or property; assert one's right or title to	seek alms, as for religious purposes
	express the need or desire for
	bark with prolonged noises, of dogs
	search the trail of (game)
	make a search (for)