I've tried to plant the wheat seeds I've won on my first croop festival many times. But it doesn't work. When I try I only get a notification that it's not possible. I've tried it on a regular and on a soil field.
I'm playing the japanese version. Don't know, if it's important for the question.
Wheat's a year-round dirt crop so it shouldn't matter what season you're in. But the notification you're receiving pops up whenever you try to plant a crop out of season, like trying to plant tomatoes in winter. So all I can say is be absolutely certain it's wheat you're trying to plant. Stupid suggestion, yes, but that's the only one I can think of.