You did not say which type of begonia you transferred to the greenhouse bed. Tuberous begonias need a resting period (no water) in the winter for best bloom. Fibrous begonias and most others are usually propagated from cuttings. Begonia seed are extremely small and since so many are hybrids may not come true or not come up at all. It is unlikely that you will get babies from seed. Again, knowing which begonia you have would be very helpful. Do you have a picture of the bed with them in it? If so, get back to us with that and we shall go from there.