quarter horse vs riding horse

quarter horse and riding horse both are nouns.

Word	Noun	Adjective	Verb	Adverb
quarter horse	Yes	No	No	No
riding horse	Yes	No	No	No

As nouns, riding horse is a hypernym of quarter horse; that is, riding horse is a word with a broader meaning than quarter horse:

quarter horse: a small powerful horse originally bred for sprinting in quarter-mile races in Virginia
riding horse: a lightweight horse kept for riding only

Other hypernyms of quarter horse include mount, saddle horse.

quarter horse (noun)	riding horse (noun)
a small powerful horse originally bred for sprinting in quarter-mile races in Virginia	a lightweight horse kept for riding only