Nobody discovered the moon. Long before there were people on Earth, there has been a moon. As a regular feature of the night sky, people must have always been aware of it.
The idea of discovering the moon implies that it had to be found either by design or by accident. Nobody had to discover the moon. It was readily visible to anyone who looked up at the sky. This doesn't mean that the earliest humans understood what the moon was. They may not have even understood that the moon in its different phases were actually different aspects of the same body.