*Both of course, are Paramount (and its fore-running TV company, Desilu Productions) franchises based on TV shows from the 1960s.

*Simon Pegg has recurring supporting roles in both the Kelvin universe Trek films as Scotty and the M:I films (since the third one).

*Leonard Nimoy's first major acting gig after Star Trek was cancelled in 1969 was on Mission: Impossible

*William Shatner and George Takei both guest starred on the original Mission: Impossible series:
https://en.wikipedia.org/wiki/List_of_Mission:_Impossible_guest_stars_(N%E2%80%93Z)

*Michael Giacchino composed the score for the third and fourth M:I movies as well as the Kelvin-verse ST movies.

*Both had movies in their respective franchises that were at one point or another, directed by JJ Abrams