If you thought Paradox's grand strategy game Stellaris sounded very Star Trekky with its big alien alliances and deep space exploration, you are not alone.
Star Trek: Infinities is an ambitious mod project that is overhauling the base game's factions and ships with those from Star Trek. It's a good fit! I guess Mass Effect could work too, but when you look at how Stellaris' universe works and what you have to do in the game, it's no wonder a Star Trek mod was one of the first things people thought of.
Here's a trailer from last year, when things were very early:
And here's a more recent playthrough from GB Gaming showing how it's come along since:
It's looking good so far -- so good that it won an award from ModDB last year -- and just added a bunch of Klingon stuff to the game. If you want to keep track of it as it goes along (it's not ready for mass downloads yet) you can do that here.