He is known to have performed before her so she knew who he was. It is however, so unlikely as to be ridiculous that they exchanged even two words of conversation. Queens did not speak to players.
Elizabeth was never Shakespeare's patron although she paid for some of his works to be performed at court (the place) for her. She did this also with many other playwrights. His company was called...
She may have met him, since many of his plays were performed during her reign, and a few under special circumstances were performed for her in particular. She could not have known him well, and...
They almost certainly met as he both wrote and performed plays for her at her palace. There is however no record of a meeting and it is possible she did not consider meeting with the entertainment....