My friends have done their courses in Ontario while living in Quebec, I don't see why it would not work the other way around. I would ask your instructor what the process would be when they mail in your "Success certificate" (a card they sent me when I did my CFSC). Chances are you'll have to see your own CFO, and they will recognize that card as valid training.
What I am unsure about is if you need to do the Bill 9 competency test if doing it through Quebec. If you do, it's usually around $45, not including rental + ammo, and I'd bring my own eyes/ears.