Final answer:
Montresor claims to be a member of the Freemasons, a fraternal organization known for its secret ceremonies and influential role during the Enlightenment.
Step-by-step explanation:
The fraternal organization that Montresor claims to be a member of is the Freemasons. This claim is made during a scene in Edgar Allan Poe's famous short story The Cask of Amontillado, where Montresor is interacting with his victim, Fortunato. The Freemasons are known for their secretive nature and elaborate ceremonies, ideals that fit well with the themes of secrecy and ritual in the story. Freemasonry was particularly influential during the Enlightenment, promoting principles of inquiry and tolerance, and had a significant impact on Western society. Notable historical figures, such as Benjamin Franklin, were members of this fraternal society.