Montreal is awesome during the summer, honestly. The city is just so ALIVE
Depending on when you are visiting, there is tons of festivals during the summer, from Jazz Festival (June 28-July 7), International Circus Arts (August), Osheaga in August (music like the Lollapaloosa one, but smaller), Hot Air Balloon Festival in St-Jean-sur-Richelieu in August too, just to name a few.
As it was said, the Notre Dame Basilic is really nice to see. There is also the Mont Royal to walk, Old Montreal with tons of shops and places to eat, the Olympic Stadium, La Ronde for the Rollercoasters, etc
There is also the International Fireworks competition being held one night everyweek if I am right. They take place at La Ronde but you can walk on the sidewalks on the Jacques Cartier Bridge and enjoy them from there.
For hiking, there is a few nice mountains in the area between Montréal and the US lines, but if you are coming from New England, they might look pretty smaller
