Because there is a theoretically finite body of "general knowledge" out there, it is inevitable that quiz shows will repeat questions. Because there are so few quiz shows on television, it is unusual when 2 shows duplicate questions on episodes airing so closely together.
From Who Wants to be a Millionaire, Friday, April 28, 2006:
(For $25,000) The registration number NCC-1701 is displayed on the exterior of what fictional vehicle?
A: Herbie the Love Bug B: Batmobile
C: USS Enterprise D: Millenium Falcon
From Jeopardy!, Tuesday, May 2, 2006:
(In the category PSEUDO MOVIE RATINGS, for $1000) It was the alphanumeric serial number for the Starship Enterprise that first appeared on TV in 1966
The same fact, but the Millionaire question is one of recognition, and the Jeopardy! one is a question of recall. The J! question is harder, but would have gotten a player only 4% of what the Millionaire one would have. Most interestingly, the contestant on Millionaire opted to keep his $16,000, rather than risk it (wisely, since he would have guessed wrong with "D"), while at least 2 of the 3 players on Jeopardy! knew the answer.
(Although Matt technically gave a wrong response of "NC-1701", I'm giving him the benefit of the doubt and assuming he was confused by the category, which required that each response "include a movie rating". NCC-1701 only kinda includes the movie rating "NC-17". Good question, bad category.)
In the 18th series of Jeopardy! (the one currently airing in reruns on GSN) there was once a category about Star Trek for which the $1000 clue involved the Ferengi Rules of Acquisition. A Trekkie was playing that day, and he got the clue correct. I'm all for including real trivia on the show once in a while, and I'm glad I saw that episode.
My thanks to the unofficial Millionaire message board at ABC7.com and the j-archive for question transcripts.