What would cause car to lose a quart of coolant in a week?

There are several possibilities for a coolant loss. There may be a leak at a gasket, hose, radiator, or water pump. . A small leak such as what you describe could be a little harder to detect as you may not see any visible dripping. Start by checking the hose connections at radiator and don't forget coolant lines to heater core. then move on to thermostat housing, check around the water pump. if the seal is starting to go on the water pump it can create a small drip. You're not experiencing a lot of coolant lose at this time so it may be possible there is a small pin hole in the radiator. This could allow coolant loss but not always give a visible sign of leaking. A radiator shop can perform a pressure test if you can't find anything.