'during the winter' is correct and usable in written English. You can use it to refer to a specific season or period of time within the year. For example: "During the winter, the temperatures in our town often drop below freezing.".
Both visited South Korea last month during the Winter Olympics.
She disappeared during the winter.
And during the winter?
"They plow my sidewalk during the winter".
Can't we go away during the winter?
Toronto has a strict uniform during the winter months.
Acqua alta occurs regularly during the winter months.
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Cristina Valenza
Retail Lead Linguist @ Apple Inc.