pilgrimage vs haj

pilgrimage and haj both are nouns.

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

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

pilgrimage: a journey to a sacred place
haj: 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, hajj.

pilgrimage (noun)	haj (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