I am going to suggest that if you can afford it, you book a room for at least one night at the Chateau Frontenac. It is amazing and when almost everyone thinks of Quebec City, the Chateau is what they picture. This is not an inexpensive place to stay. You will pay a minimum of $250 for a room and most likely a lot more. It is however, worth it for the experience. It is what we did, spend one night and I will never forget it. They brought Alex cookies and milk, obviously he liked it.
If what you want is the location, the Hotel Bellevue is literally in the shadow of the Frontenac and it cost less than half the price. It is a lovely small hotel with a wide variety of room styles. It is located within the walls of the city.
If the lower city appeals to you, the Auberge St. Pierre is located across the street from the rear entrance to the Museum of Civilization. There is plenty to do in the lower city and this is a quiet and interesting area to stay. Breakfast is included and this is a very charming place to stay.
The Relais Charles-Alexandre is located outside the walls of the city on Grande Allee , which is the Champs Elysee of Quebec. It is within walking distance of a variety of great restaurants on the Rue Cartier and if you enjoy walking, you can walk into the old city. It is a boutique bed and breakfast hotel with classic French Canadian décor. It was a very comfortable bed as Al is happy to show.
The Hotel Manoir Victoria is another of our favorites. It has something quite unusual in Quebec, an indoor swimming pool. It is one that we liked enough to stay more than one time. It is located in a great shopping area.
Other hotels we stayed at more than one time, include the Lowes which is being converted into senior housing and the Delta which was not a Delta when we stayed twice so I decided not to mention them.
We have personally stayed in all of these hotels and would recommend them as excellent options. Check out their websites, it really depends on how much you want to spend and what you plan to do. None of these would be considered a bargain, for that we stayed at the Super 8 which is really nice, had a great indoor pool with slide but is located about 15 minutes outside the city.