Your player name in Star Trek: Legends is the same as your Game Center name. So it cannot be changed in-game.
To change it outside of the game, follow these steps (but keep in mind that changing your Game Center name means it will be changed for all titles that use it):
- Go to the App Store
- Tap/click the account icon in the upper right
- Choose "Game Center" (second down in the list)
- Tap "Nickname" and change to what you'd like before tapping/clicking "Done"
There may be a delay between changing your account name and having it appear in-game.