He died on 15 Aug 1852 in Ste. Anne.
I checked for birth and/or death information at FindAGrave, and using Jacques' and his parent's names to search in Family Search. I was unable to connect with ancestry.com at all. I looked at Google for a few minutes, but came up empty. A MyHeritage entry for the family indicated that Lambert and Mary had only two children, and it didn't appear that Jacques was one of them. Omission on the part of that site, I expect - I'm not saying he never existed.
When checking under his father's name, I used both Louis" and "Lambert", as well as each one as given name, as it became apparent that he went by "Lambert". I looked at some of his siblings' names to see if there were any clues, and while they might be there, I was unable to discern any. May be worth another look.
I looked at newspapers.com to see if there was any birth or death notice for young Jacques, using both Pitre and Peters, but found none.
I did find a death record for his father. It was recorded under "Lewis Peters". That source had already been identified on Louis Lambert's profile.
I also found a real estate ad that appeared to be selling off part of an estate once known as "Lambert Beaubien farm, the second part of that figuring into maternal ancestry. (Jacques' maternal grandmother's full main name included "dit Beaubien" suggesting a possible abduction by Indians, sale to the French, etc.) The reference may have been, instead, simply using the farm as a landmark to tell where the land being sold was located. Curious how "Lambert" and "Beaubien" name the property, unless it was named for the couple specifically. Otherwise, it's possible they were cousins before they were a couple. When I was looking at FindAGrave, I found one child already linked, then found two more to link to the couple by clicking on the first Find Others link on the right -- in the same cemetery. I did NOT follow up with looking at all possibilities in the city or in the county. Perhaps Jacques was buried on the farm.
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Jacques Francois is 19 degrees from Vinnie Hoxie, 19 degrees from Frederic Remington, 23 degrees from Pablo Picasso, 18 degrees from Edgar Degas, 22 degrees from Alexander Calder, 24 degrees from Camille Claudel, 20 degrees from John Bacon, 21 degrees from Barbara Hepworth, 25 degrees from Norman Lindsay, 23 degrees from Frances Loring, 22 degrees from Florence Wyle and 24 degrees from Elaine Weatherall on our single family tree. Login to find your connection.