dormitory vs quarters

dormitory and quarters both are nouns.

Word	Noun	Adjective	Verb	Adverb
dormitory	Yes	No	No	No
quarters	Yes	No	No	No

As nouns, quarters is a hypernym of dormitory; that is, quarters is a word with a broader meaning than dormitory:

dormitory: a college or university building containing living quarters for students
quarters: housing available for people to live in

Other hypernyms of dormitory include building, edifice, living quarters.

dormitory (noun)	quarters (noun)
a college or university building containing living quarters for students	housing available for people to live in
a large sleeping room containing several beds