We don't know because the listener is never identified, but the listener's identity is not important to the story. To whom Montressor is speaking is irrelevant. The significant element is that Montressor tells his story himself, thus revealing his character. As Montressor recounts what he had done so many years before, his madness becomes chillingly apparent and the horror of the story's conclusion is realized. The first-person point of view is most effective in creating the story's tone.
This seems to be an end of life confession, perhaps made in writing rather than spoken. He asks for no forgiveness and states that for half a century no mortal has disturbed Fortunato's bones.En Pace Requiescat which means rest in peace.
From this we can determine that Montresor committed the murder at least fifty years earlier. At the time of the murder, Montresor has had time to rise in prominence and then lose the respect of others ("you are admired as I once was"). It would seem that at the very least, Montresor is 75-80 years old, if not more. That makes it seem likely that it is an end of life confession.