This is 200 year old history, the church isn't responding because they have no more data for you. There isn't another record that names his mother, that is the only relationship that can be possibly determined since the parents weren't married. They were most likely in the community, maybe even related in some manner to the one or both of the godparents.
"inconnu" means unknown = the child was born out of wedlock.
There is nothing in the record that indicates the infant was First Nation, that's your first clue the family lore might be nothing more than lore. Trying to prove family lore distracts from finding the facts.
What surname did Eusebe adopt? Did he marry in Quebec? Do you have that record? Are any guardians or parents cited on it?
Was Josephte Fonteneau married at the time of his birth/bap? What was her age? Did she have female siblings? If she was his guardian, she's the focus of your research.
Families tended to take care of their own. I would research the Fonteneau line, and see if you can come up with some possibilities. The best you can expect is to narrow down maternal candidates, the paternity is not possible. Look for a Eusebe in the Fonteneau line in the previous generation.