First thing you need to know is in Quebec you have the National Parks like everywhere in Canada (managed by the federal government), and then you have the Quebec National Parks. These used to be the Quebec Provincial Parks (like every other province has it's provincial parks) but have been renamed "National" parks, for political reasons... These National ex-provincial parks are managed by the provincial government, via a government-owned organisation named SEPAQ (Societé des Établissements de Plein-Air du Québec).
So everybody is confused because both the federal and provincial parks in Quebec are called "National Parks". In all other provinces, provincial parks are called just that.
So here you have the National Parks of Canada: Parks Canada - Home
You will see that there aren't too many of these parks in QB, the one just north of Trois-Rivière is Mauricie NP. It's a beautiful park, and if you like camping in parks you will love it.
Here you have the National Parks of Quebec: Sépaq - Société des établissements de plein air du Québec
They are very nice parks as well, on par with the federal ones.
Now the dogs. You will find the pet policies on their respective website.
Generally, the federal parks allow dogs in campgrounds, but not on hiking trails, beaches, etc.
In Quebec parks (provincial), traditionally dogs have never been allowed anywhere, but this is about to change, as next summer dogs will now allowed in a couple specific parks on a trial basis. Check their website for details.
Be aware that it can be difficult to get a site during weekends, so make reservations early. Also, don't forget your bug repellent. Early in the summer is the worst time for the mosquitoes and black flies, they can be extremely aggressive and ruin your stay.
I allow myself to make a couple suggestions. There is a very nice little regional park just east of Trois-Rivière that's a little jewel, the Batiscan River Park:
Parc de la riviÃ¨re Batiscan - Parc RÃ©gional et Via Ferrata
You mentioned Montreal and Trois-Rivieres, I don't know if you have ever been to Quebec City. If not, you absolutely have to visit Quebec City. It's nothing like Montreal. Québec City and Area : Official Web Site - Québec City Tourism
I can suggest a couple nice campgrounds near QB City if you like.
Also, if you're looking at a campground elsewhere and not sure if it's a good choice or not, send me a message and I'll give you my opinion on it. Most private campgrounds allow pets.
You're from NC, we've been to the OBX a few times (Natl Sea Shore campgrounds) we enjoyed it a lot.