four-footed vs quadruped

four-footed is not a noun while quadruped is a noun.

Word	Noun	Adjective	Verb	Adverb
four-footed	No	Yes	No	No
quadruped	Yes	Yes	No	No

As adjectives, four-footed and quadruped are synonyms defined as:

Other synonyms of four-footed include quadrupedal.

four-footed (noun)	quadruped (noun)
	an animal especially a mammal having four limbs specialized for walking

four-footed (adjective)	quadruped (adjective)
having four feet	having four feet