quest vs lay claim

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

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

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

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

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

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