It is perfectly possible to grow amaryllis from seeds if you don't mind waiting quite a long time for new flowering plants. You must also not expect them to come true to the variety which produced the seed, though you may be lucky.
Seeds should be treated in the same way as lily seeds though the plants should be kept frost free during the two or three years they will need to reach flowering size.
I'd say it would be much easier to remove new 'bulbs' from the main amaryllis bulb and grow this on to make a new plant, as this will take far less time than seed and can be guaranteed to produce the same flowers.