To maximize heat in winter, should I remove my window quilt during the day?

If you ask this question it probably means that the structure is thermally insufficient, and thus the existing heat leaks dwarf any gains from playing with window coverings that only modify solar heating - unless you have lots of windows and lots of Southern exposure.

In my experience, when such questions get asked things are pretty bad, and the only actions to be done on the building that will really change things appreciably are: investigate how to improve the thermal efficiency of the building if you own it (including window, wall insulation, trees to shield from wind, etc.), or move if you rent. Everything else probably takes more resources to just talk about than any savings you'll get, although for sure there can be plenty of placebo effect.

The one thing that will actually make the biggest difference right this very moment is to wear warmer clothes indoors and let the temperature of the room be lower. It literally has the highest cost-to-benefit ratio that you can get in cold interiors, and thus must be investigated first. That's the only way to save significant amounts of energy and to improve your winter comfort by a lot in thermally insufficient buildings if moving or altering the building is not an option. You can always set the thermostat higher if you are OK with the added expense.