One thing though - the alien captain had certainly heard of it.
He may have heard
of Genesis, but there's no way of knowing how much he knew about the planet or the project that spawned it. That he knew it wasn't allowed could be for any number of reasons. Maybe his ship makes regular cargo runs near the Mutara sector and was stopped by Grissom
or another Federation vessel patrolling the area.
He knew who Dr. McCoy was, but the Enterprise
crew was the Starfleet equivalent of the Beatles. I don't think there was anyone who didn't know who Kirk and his friends were. Even Mr. Adventure was in awe of the admiral when he strolled into the transporter room later that evening.