No family name is 'natural' as that would imply that it was passed on genetically. All family names are 'artificial'in that they've been passed down consciously by someone, usually the parents, even if they don't understand the reason behind the family name.
You'll notice that many English family names are names of trades of professions, towns and cities, or even personal names with the suffix -son attached to them. This is because historically the family name was not passed down from parent to child as it is today, but rather, depending on the tradition, the child's city of birth, the father's name, or later in life according to his trade or profession, bearing in mind that a child's trade or profession often followed in his father's footsteps. So the idea of the father's given name forming the child's family name is certainly not foreign to English culture.
Since you're in Quebec, we have to consider French culture too. In French culture too though you'll notice many family names formed from the names of cities, trades or professions, owing to a similar history. So your family tradition is not so foreign to French culture either. Many cultures around the world continue to follow the tradition of the child's second name being the father's name, and the third name being the paternal grandfather's name, the name of his birth town, or the name of his tribe. Often in the same town various family traditions are followed in this regard and sometimes they are even interchangeable. None of this if foreign to historical French and English cultures, and so I'm sure the Ministry will likely accept this without difficulty; but confirm this with the Ministry anyway just to be sure, as I may be wrong.
One thing I do know is that the law generally likes consistency, so my guess is that they'll certainly accept if not even encourage or even require that you follow the standard in your passports.
