Quail are hardy little birds. They are compact making them able to survive in colder winter temperatures. There isn't a lot that you need to do to care for them differently than chickens. Raising quail during winter is about keeping them sheltered rather than keeping them warm.
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Raising Quail During Winter
I first want to say that our temperatures in Southeast Nova Scotia don't get that cold. And that's where we live, and I get all my knowledge of bird raising from. The temperatures in Nova Scotia can drop to -29 °C (-20 °F) in the winter, but it's not common.
Housing Quail During Winter
You need to provide shelter from the elements. Here we get snow and rain all winter, sometimes on the same day, and sometimes at the same time because Nova Scotia weather can be crazy.
I like to give them more shavings, and sometimes smaller boxes in their cages to hide in. It can be a box, with shavings in it, often its fur tree branches. Quail like to hide and these provide a space for them to hide.
My summer watering system isn't going to work for the whole of winter, so I have to set something different up. They need fresh water, and in the winter you might have to get it for them more often than you are used to.
Of course, they need feed still. Quail need high protein to lay eggs. One other thing I give is scratch grains which are high fat, and good to give in the late afternoon to help keep them warmer at the night. This does not replace their wild game feed, or whatever it is you are feeding your quail.
Indoors VS Outdoors
I leave my birds outside during the winter and let them freeload. I could bring them into the basement and give them supplemental light to get them to continue to lay eggs I don't want them in my basement so they get the winter off.