No, Scottish actor Martin Quinn is not related to the famous Desilu TV producer Quinn Martin. Their names are a mere (astonishing) coincidence.
Scotty was, of course, the chief engineer on board the U.S.S. Enterprise from the start of the series in 1966 and last appeared in “Star Trek: Generations” in 1994. Scotty also appeared on an episode of “Star Trek: The Next Generation,” having been kept alive in a transporter buffer for many years. Throughout, Scotty was played by actor James Doohan. In 2009, an alternate-universe version of Scotty was played by Simon Pegg.
Here’s a fun piece of trivia (culled from the 1997 book “Inside Star Trek: The Real Story”): Gene Roddenberry nearly cut Scotty from the series, thinking that the Enterprise didn’t require an engineer. It wasn’t until Doohan’s agent confronted Roddenberry that Scotty was allowed to remain a part of the show. Doohan has told the tale numerous times at conventions, but he was prepared to give his character one of numerous dialects. He nearly made Scotty Russian, as he could do a good Russian accent. Eventually, Doohan decided that his character should be a Scot, as he knew several clever Scottish engineers in the War (Doohan fought at D-Day).
Quinn is the first actor of actual Scottish descent to play Montgomery Scott. He is currently 28 years old, although Doohan was already 44 at the start of Trek. In “Hegemony,” Scotty is able to build, from scratch, a lantern-like device that lures Gorn monsters into a trap. When he is rescued from a desperate situation and brought on board the Enterprise (for the first time!), he is recognized by Pelia (Carol Kane) a former teacher of his. Evidently, Scotty was a brilliant engineer who got terrible grades.