Poe does not tell us directly, but only implies it in the third paragraph (p 135, paragraph 3):

"...but in the matter of old wines he was sincere. In this respect I did not differ from him materially;—I was skillful in the Italian vintages myself, and bought largely whenever I could."

Poe reveals a potential source of rivalry or competition between the two: their expertise in wine. Fortunato probably insulted Montresor's knowledge of vintage Italian wines. By insulting Montresor, Fortunato has insulted Montresor's family name and the French race.