Some of the previous posters have speculated that you are interested in "Vieux Montreal" or the old part of the town. The term "Latin Quarter" is also sometimes used to refer to the area around the "Gay Village" that runs along Ste Catherine Street East.

Regardless of your desired location, the temperatures will be chilly enough that I don't know that you will want to "stroll" very much. But you can certainly visit. I have been to Montreal many times in January, and find it a beautiful city any time of the year.

One thing for a non-native to watch our for in the winter - they remove snow from the sidewalks with these small bulldozers. City workers drive these things around on the sidewalks like they are on a Formula 1 track. It is a bit of an exaggeration to say that pedestrians are diving for cover when one comes down the sidewalk, but it is certainly a good idea to pay attention when you hear one coming.