Your CO alarm is going off. Do this first.
Leave the house. Everyone, including pets, into fresh air. Do that before you read the rest of this.
Then call 911 or your gas utility's emergency line from outside. Most utilities send someone within an hour and it's usually free for a CO or gas call.
A carbon monoxide alarm on a gas furnace in winter is an emergency, full stop. CO is a byproduct of burning gas, and a furnace that's venting it into your living space can be lethal in a few hours.
You don't need to figure out the cause right now. You need to get out and get help. The diagnosis happens after the air is clear and a pro has a meter on it.
Why this happens in winter specifically
Your furnace works hardest on the coldest nights. Long run cycles mean more combustion, more exhaust, and more chances for a small venting problem to become a dangerous one.
Cold also seals up houses. Windows stay shut, and modern homes are tight. CO that would have leaked away in October now builds up.
Snow and ice do their part too. A sidewall vent (common on 90%+ high-efficiency furnaces) can pack over with snow and choke off the exhaust path.
And heat exchanger cracks behave worse when it's cold. The metal expands more from a hard cold start, which can spread a crack that stayed hidden in mild weather.