There are some suggestions, but they’re all in reference to the Long Night, so it’s not clear whether that includes lesser winters:
“Fear is for the winter, my little lord, when the snows fall a hundred feet deep and the ice wind comes howling out of the north. Fear is for the long night, when the sun hides its face for years at a time, and little children are born and live and die all in darkness while the direwolves grow gaunt and hungry, and the white walkers move through the woods.“
“..The Others,” Old Nan agreed. “Thousands and thousands of years ago, a winter fell that was cold and hard and endless beyond all memory of man. There came a night that lasted a generation, and kings shivered and died in their castles even as the swineherds in their hovels. Women smothered their children rather than see them starve, and cried, and felt their tears freeze on their cheeks.” Her voice and her needles fell silent, and she glanced up at Bran with pale, filmy eyes and asked, “So, child. This is the sort of story you like?”