No you don't need a car in Quebec City.
There are flights to QC from Montreal pretty much every hour on Air Canada, or you can take the train (or bus).
If you intend to do any activities outside the city, a car is handy to have, and you can always just rent one for the day.
Do you have winter driving experience? You have not included your location in your profile, which is always helpful in these situations for giving advice.
If you stay inside the gates in Old Quebec there is no need for a car. Look for a hotel in Upper town - Chateau Frontenac is the reference point to how far things are.
Quebec City airport is small .. Montreal is the larger hub. It entirely depends on what works for you.
Bus: https://www.orleansexpress.com/en/ Note that bus service has more departures from each city. It is not like Greyhound in the US ... much better facilities and service - often used by business people and travelers.
Air Canada: https://www.aircanada.com/ca/en/aco/home.htmlEdited: 23 November 2017, 05:00
If you can get a direct flight, go for it. YQB is a smaller airport, so shorter lineups etc. make for a more pleasant experience.
If not, there are trains from Dorval to Quebec. Dorval is right beside YUL and there is a free shuttle, which makes for a relatively quick transfer. If you take a train/bus from Montreal (downtown), you have to add at least an hour for the transfer, closer to 2 during rush hour.
If you've been up in the Appalachians, you'll be fine. It's the greasy, slick, semi-frozen stuff that is the worst. That and finding parking! Montreal & Quebec city were built before cars, so a car is a drawback.
Bus: Orléans Expres, some of their routes starts at the airport, the others from Gare d'Autocars de Montréal which is in the Latin Quarter and the terminus of the airport shuttle (747) and within the flat rate zone from the airport for taxis.
Train: Via Rail, some trains will make a stop at Dorval, others will leave from Central Station which is downtown within the same perimeter for the 747 bus and cabs re the rates and accessibility.
Depending on how long you have it might be fine to add a few days in Montréal too. In fact you could just spend one night / 2 days in Qc and possibly the night of arrival and before departure in Montréal.