Parc de la J-C is around 20-30 minutes north of Quebec city. As a national park, they let us ski in the wood in two different area but they don't cut any tree. So the area is not wide open... there's lot of tree and branch but there still so good ski to do if you want to do some backcountry close to Quebec city. Not really high (probably 250 meters?) but you can do multiple laps. The park open the ski area when there's 1 meter of snow on the ground. If you plan to visit Quebec city, bring your ski and you will have fun but I will say: don't plan a backcountry ski trip in Quebec !
If you plan a trip to Quebec, you can also do some slack country at the Massif de la petite rivière St-François. Good ski to do but you will have to pay for the lift ticket... expensive day but worth the view!
Don't miss a trip to Gaspesie if you plan a trip in eastern Canada. Around 6h east of Quebec city. The montains are called the Chic-Choc: you will have more snow, alpine bowl, more remote skiing... Really good backcountry. Bring you're avalanche gear for this place. Some people will complain that it's not like skiing in the west... yeah sure! But I ski in Utah, many resort in British Columbia and Alberta, backcountry in Roger Pass.... And I still remember some awesome skiing day in Gaspesie ! So maybe not as high but still lots of good turn to do !
Hope I help you to plan your trip !