When Moe describes his "Time Machine," he says "Suppose it's Friday at eight, and you want to go back to Wednesday at ten because you missed your favorite TV show." This was something of an in-joke as at the start of the season NBC had moved the show's time slot from Wednesday at 10 (Eastern) to Friday at 8 (Eastern) and the show's producers were lobbying to return the show to its original time slot.
The calculating device that Captain Galaxy uses on his show is very similar to the device that Al uses throughout the series. Since Sam Beckett watched Captain Galaxy as a child, it can be assumed that he was inspired by Captain Galaxy for the design and possibly other ideas.
As he exits the "Time Patrol" time machine prop, Sam trips and sprains his ankle. This was written into the script to account for the injury to his foot that Scott Bakula sustained during the filming of the earlier episode Runaway - July 4, 1964 (1991).
The time period of this episode partially overlaps with Lee Harvey Oswald - October 5, 1957 - November 22, 1963: Part 1 (1992). As with this episode, the Japan segment of that episode took place from October 6 to October 7, 1957.
Richard Herd, who starred in this episode (#3.13) as Moe Stein, also appeared on Seinfeld (#6.18) as George's boss Mr. Wilhelm. In one of the episodes (#4.3) of Seinfeld, there was a Quantum Leap poster in the background of the NBC waiting room where Jerry and George were sitting before pitching the idea for Jerry's sitcom. Additionally, several Seinfeld side characters also appeared on Quantum Leap including Patrick Warburton, Louis Mustillo, and Valerie Mahaffey to name a few. Richard also appeared as Admiral Paris in Star Trek: Voyager (1995) which make him and Scott Bakula two actors who appear in this episode to later appear in the Star Trek franchise.