Normally the free form outdoor skating rink at Place d'Youville should be open as well as the tobogan at the end of Terrasse Dufferin. If you want to do an activity out of town then Village Vacances Val Cartier will be a load of fun for quite a few hours.
Alll the ski slopes around have snowmaking machines thus should be open. The largest, Mont Sainte-Anne, will also offer other activities like snowmobiling and dog sledding depending on natural snow conditions. The great plus with this ski station is that it runs bus shuttles to Old Québec so if you don't have a car or don't want to drive in bad weather it comes in handy.
And, of course, all the Christmas decorations will be up and Rue du Petit-Champlain always try to make it a White Christmas Wonderland.