in Montreal you can use the Metro (subway) and buses to get around - a car is a nuisance
to get to Quebec City there is a train - VIA Rail that does daily trips and Orleans Express Bus that does hourly departure. In QC you only get buses.
Best to get to QC first then back to Montreal so you are in Montreal the night before you flight out.
In QC - There is an Upper and Lower town connected by stairs or a funicular. Lots to do just walking around and exploring.
May or June is not the time for any hockey training session.... It's golf time for these guys!
(IF, a big IF, the CH makes it to the Stanley cup's finals and it is May, tickets will be VERY expensive.)
I would like to add that when you have definite dates, it would be good to come back here and let us know. It might be easier to inform you what will be happening at that time... May June is really vague.. unless we are suggesting indoor activities... and these are all listed in the Things to do" at the top of this page)
Take note that Montreal has a F1 Grand Prix in early June (8-10)... Hotel prices are really expensive at that time.
About the car: it is best not to rely on a car. Use public transportation while visiting the city.