I have been to both Montreal and Quebec City many times, including a couple of years ago when we went to Montreal with our then two year old as part of a longer trip. I absolutely love Quebec City, but if you have just three days, I agree with the other poster that it is not worth trying to go there as part of a day trip.
As to AirBnb/VRBO/hotel, I think that's up to you and your own travel style. Our daughter was an excellent sleeper at that age, so we usually stayed in hotels (also, most hotels have free pack and plays that they'll put in your room). But if you want to the space and to have a separate room from your child, the apartment/home rental might be a better option. We stayed on that trip at the DoubleTree, which is close to both the Vieux Port, Mont Royal, and lots of other places of interest. As well as two subway stops (along different lines). If it is your first time in Montreal, I would recommend somewhere in that area if possible - there are a lot of hotels there.
If you want to go for a nice walk, there is a long, slow incline path up to the top of Mont Royal - hint, when your child needs a nap, pop them in the stroller and enjoy. The views from the top are awesome. I would probably avoid the steep, more direct route to the top. I think there was a playground at the top too. There are also a lot of nice pedestrian only streets and outdoor markets. Our kids have enjoyed those, lots of sights, smells, treats, etc. And definitely explore the Vieux Port.