the ones grown from corms you can, yes, but not the fibrous rooted bedding ones. Remove the corms and lay them on their sides so the foliage is sideways - once that dies back, let them dry off a bit and keep them somewhere cool but frost free till early March, when you start them off again in pots indoors or in a greenhouse ready to plant out at the end of May.

I'd expect to get a month or two more of flowers from Begonias though, so I wouldn't dig them up til end of October at least.

Check the stored tubers/ corms for vine weevil grubs from time to time as they love to feed on them over winter.