Ann, you're close enough.
It's actually "Tribbles and Tribulations" an episode of Star Trek: Deep Space Nine -- which, in fact, basically puts the cast of DS9 on the Enterprise, in the past, during the events of "The Trouble with Tribbles"
...and it's got Dax swooning over Spock. hehehe! Which totally reminded me that I used to have a *huge* crush on Mr. Spock when I was 6 years old. *giggles* (I also loved cmdr. Koenig, from Space 1999 -- I guess I was a sci-fi fan real early on. haha!)
