the Phrase is sorta from Star Trek, the TV shows in the 1960's.
It's extremely famous, sorta.

Remember the line Play it Again Sam? Humphrey Bogart says it to Sam, the Piano player, in Casablanca. It's a 1940's black and white cinema classic.

Just like "Play it Again, Sam!" the line "Beam Me Up, Scotty" is never actually said.
It's one of those geeky movie/TV lines everyone knows. But was never actually said. The ship captain says 'Scotty, Beam is up.' Or, "scotty, two to up' or other versions, but never actually says the famous line.

Same with Bogie's character in Casablanca.
"Play it Sam," is as close as our dear love torn brooding patriot says!