Your profile photo in Star Trek: Legends is the same as the one you have chosen for Game Center. 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 profile photo 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)
- Choose "Edit" under the icon at the top
- You'll now either be able to create a Memoji (the + symbol on the far left) or choose from a number of pre-existing icons
There may be a delay between changing your account photo and having it appear in-game.