My wife is Québecoise, so we head up north every summer to visit her family. And we ALWAYS stop for a day or two in Québec City, on our way up to Gaspésie and sometimes on the way back home.
If you want to visit something outside of the city, there are a few "natural" attractions that you might enjoy seeing. We visited them all in one day. There is the "Parc de la Chute Mormorency" in nearby Beauport:
http://www.sepaq.com/ct/pcm/en/ It is free to walk around the park, but there is a small parking fee. You can park at the bottom of the falls and take the cable-car up to the top of the falls and back down. If not using the cable cars, there are stairways that you can use to get to the top. There are spectacular views of Québec City, the St. Lawrence River and L'Ile d'Orléans.
A little ways up the road on the way to our next destination, we stopped at a Honey Economuseum, Musée de L'Abeille".
http://www.musee-abeille.com/en/default.html It is free to visit. It has an exhibit about the history of beekeeping and a large glass beehive that you can observe the bees at work. Try to find the Queen bee, she is marked with an "X". Before exiting, you will pass through a shop that sells pastries and candy made from honey and maple syrup. And you can taste some samples of different types of honey. We have young kids and they enjoyed it.
Finally, we set off to our final destination, Canyon Ste.-Anne:
http://www.canyonste-anne.qc.ca/index.html This is a picturesque canyon that has trails, suspension bridges and many scenic overlooks.