pilgrimage vs hajj

pilgrimage and hajj both are nouns.

Word	Noun	Adjective	Verb	Adverb
pilgrimage	Yes	No	No	No
hajj	Yes	No	No	No

As nouns, hajj is a hyponym of pilgrimage; that is, hajj is a word with a more specific, narrower meaning than pilgrimage:

pilgrimage: a journey to a sacred place
hajj: the fifth pillar of Islam is a pilgrimage to Mecca during the month of Dhu al-Hijja; at least once in a lifetime a Muslim is expected to make a religious journey to Mecca and the Kaaba

Other hyponyms of pilgrimage include hadj, haj.

pilgrimage (noun)	hajj (noun)
a journey to a sacred place	the fifth pillar of Islam is a pilgrimage to Mecca during the month of Dhu al-Hijja; at least once in a lifetime a Muslim is expected to make a religious journey to Mecca and the Kaaba