In his foreword to The Physics of Star Trek, Hawking jokingly wrote that after "winning" the holodeck card game, he called Paramount Pictures to try and cash in his chips (Federation Credits), but they didn't know the exchange rate."
Stephen Hawking was on the Paramount lot for the video release of his film A Brief History of Time (1991). He asked the staff "Could you take me out of my chair and put me in the captain's chair?" His request was granted.
Stephen Hawking, while appearing as guest star, asked the crew for a tour of the Enterprise. When he saw the Warp drive he said "I'm working on that..."
When Stephen Hawking says "Wrong again Albert", its a reference to the Professor's life in physics. He's disproved some of Albert Einstein's theory.
This episode is the third time that the Enterprise-D is commanded by a female officer (Beverly Crusher). Tasha Yar was the first, while Commander Riker was supervising repairs at the holodeck in The Big Goodbye (1988). Deanna Troi was in command during Disaster (1991), as the highest ranking officer known to be alive after the ship was struck by quantum filaments.