If an alternator supplies electricity to an electric motor that spins the alternator will it run forever?

Yes, you could connect these parts together and it would operate. But after awhile you will find that the batteries are running low and the will motor is run slower than initially.

The answer to the specific question is "no - you could not do this to generate electricity endlessly". If you could, that would constitute a perpetual motion machine, and that is impossible to do, due to losses in energy conversion because nothing is perfectly efficient.