This is a Federal jurisdiction, nothing to do with Quebec.
I know when I drive to the US, I get asked how long I will be staying. So you would expect the same thing coming to Canada - proof of long-term residency in the US, short term stay here should be fine, but it is not a situation I have direct experience with.
If I were going someplace for 4 months, I would have a fair bit of stuff with me too. I had a fairly full car visiting a freind for a week (but that included the dog).
Pets? You will probably need proof of rabies vaccination, you should check that.
It depends on the border how strict they are, and it may raise a red flag if you have a lot of stuff with you. I'd check into Quebec rules just to be safe. (I went to McGill and crossed the border a lot, but it was also 10 years ago. Sometimes I got bugged even with a student visa.)