My info is kinda dated - I moved away 20 years ago. But I keep in touch with my childhood buddies, who are all avid hunters. For what it's worth, here are a few things to consider:
1. As a resident, you can hunt unguided for most things. Finding good spots on crown or private land is going to be your bigger challenge.
2. Deer - you will probably find info in the links provided above, but I believe that rifle season is still quite short - two weeks with the weekends on either end (three weekends, two weeks), but there is also a blackpowder, crossbow, and bow season (but only one tag I think). I'm not sure if the seasons overlap or how that works. Bucks only on your tag, and you apply for region-specific doe tags.
3. Caribou - as a resident of Quebec you can go up the highway past Mistassini and get on the transtaiga highway (dirt) that runs along the south side of the La Grande Riviere (where hydro quebec built the big dams). There are caribou hunts to be had up there. I'm not sure the details, but I know that it can be done unguided.
4. Unlike some places (like Nova Scotia), non-residents of Quebec can hunt unguided (tags cost more though), so when you first get there, even if you're not all set up officially as a resident, you can buy a tag and go out hunting (if you can find a spot).
5. One option that you can take advantage of that falls somewhere between the guided/outfitter and self-guided options are the SEPAQ hunts in their provincial Reserves. There is one just about 90 minutes north of Quebec City (Laurentides Reserve). Moose tags are by draw (it's already happened for 2015). They do fully guided hunts, but they also do "European Plan", so if you get drawn you choose a zone and time based on what order you were drawn in (first name drawn gets pick of the litter, and so on). The hunts range in price from the fully guided American Plan to camping. But you do get exclusive access to the hunting area during your period of hunting. Even the European plan is not cheap, but it's also great for folks who don't have access to private land or good crown land, or who dont' have a camp. You have to be a Quebec resident to apply, but if you win you can invite non-residents. My buddies from Quebec invited me last fall in the Laurentides reserve. We had a beautiful cabin included with our hunt on a massive territory. We shot two bulls (6 hunters). Nice logistics such as access to a cooler to hang your quarters in while you hunt to fill your second tag or while you just hang out after the hunt for the duration of your stay. I believe it cost somewhere between 6000 and 7000$ for the six of us, which included the cabin and access to the territory. You have to buy your own tag. Group size is minimum 2 hunters per tag, and a maximum of 4. There are bow tags, muzzleloader tags, rifle tags etc.... that you can apply for. Prices vary - this year we got drawn again and it's more expensive in a different Reserve. We will be hunting in Matane, two tags and a cabin for 6 hunters was 10,000$. So as I said - not cheap, but it's an option. They also have day rates for grouse hunting with or without overnight stays in the cabins, which you don't have to apply for and might be a nice way for you to scratch the hunting itch when you first get there.
As is often the case - extensive and thorough scouting using all the tools at your disposal, and forming/nurturing relationships with other hunters and landowners are the key to carving out your hunting options. But these things take time - in Quebec as they do anywhere else!
GOOD LUCK!
