Problem: The snow doesn't melt until June and is back in September. Short running season.
Really, though, in a quick maps.google.com search I found ONE school with a running track in all of greater Quebec City: De Rochebelle High School. You can search maps.google.com, or just google.com, to get its precise location. Click on the "satellite" rectangle at lower left to get imagery, and you'll see a 400m running track. Doesn't look to be a very good surface, but it's the only track I've found in my limited search.
At the upper left, I hit an arrow to expand a search box, then I searched on "high schools in quebec". I was quick about it, but I could only find that one running track mentioned above! High schools there don't have tracks around their soccer fields.
If you scour that satellite image, back and forth, through the city maybe you'll find another track, like in a municipal park, not a school's track.
There must be a local running group in Quebec City, found hopefully in a google search. I'd guess that someone would be pleased to point you to several options.