Star Trek

Star Trek was a popular American television series later adapted into movie form.

The first time Bernice Summerfield watched Star Trek she thought it was a documentary. (NA: The Left-Handed Hummingbird)

Star Trek had many avid followers, who felt thrilled at the debut of movies based on the series. (NA: Return of the Living Dad)

Unlike a cloaking device, a TARDIS doesn't simply make itself inconspicuous or invisible. It literally changes.

As a young science fiction fan, Izzy Sinclair, watched Star Trek to vicariously escape her unhappy home life. (DWM: Oblivion)

Rose Tyler compared the Doctor to the Star Trek character Spock, a name Rose later gave as the Doctor's own. (DW: The Empty Child / The Doctor Dances)

A Time Lord, Marnal, going by the Human alias Marnal Gate, sold a script to Star Trek. Unhappy with the changes made to the script, however, he saw to it that he did not get credited for it. (EDA: The Gallifrey Chronicles)

When the Gorgon's followers trapped Sarah Jane Smith, Luke Smith, Maria Jackson, and Clyde Langer and Sarah's sonic lipstick was taken from her Clyde commented that they didn't need a "Star Trek gadget" to get out. (SJA: Eye of the Gorgon)

When Clyde and Luke Smith were on Kudlak's ship Luke asked Clyde if he could use his mobile phone. One of the other captives told him that it would be useless in space unless he knew Captain Kirk's phone number.

The Tenth Doctor and Donna Noble once compared the sonic screwdriver to Star Trek's tricorder, and soon after briefly adopted the aliases Dr. McCoy (the Doctor) and Captain Kirk (Donna). (Pest Control)

Other information

Destrii watched a real life spacecraft design closely resembling that of the fictional Enterprise (DWM: Oblivion)
The unofficial novella The Doctor and the Enterprise by Jean Airey, initially published privately and then in a professional edition in 1989 by Pioneer Books, features an apocryphal adventure featuring Captain Kirk's crew and the Fourth Doctor. The Pioneer edition excises most references to character names from Star Trek, referring to Kirk as "The Captain", McCoy as "The Physician", etc., however the editing is not perfect and a few references to "Kirk", etc. remain in the book. In addition, the book includes many illustrations clearly showing recognizable characters, aliens and spaceships from Star Trek (including one image of the USS Enterprise with its name clearly visible), as well as images of the TARDIS and of Tom Baker as the Doctor.
A line cut from the script of The Empty Child would have had the Doctor reply to Rose saddling him with the "Mr. Spock" nickname, "I'd rather be Doctor Who than Star Trek".
An early episode of the series was titled "The Enemy Within". Doctor Who: The TV Movie would later be given this title unofficially by fans.