There are multiple ways in which you can compete directly against other Captains in Star Trek Timelines.
The Battle Arena is a place to test your tactics in commanding a Ship, by pitting you against simulations of other Captains and their battle configurations. You can work your way up in these persistent Leaderboards to gradually earn additional Ship Schematics and assert yourself as a foremost expert in space combat. You can read more about the Battle Arena here.
You can also throw your lot in against other Captains in Q's brutal yet amusing Gauntlet, where your Crew's Skills will be tested against others to determine who among you is best equipped to handle the current multi-dimensional crisis. Beware, though, that Q's rules tend to be erratic and unpredictable. You can read more about the Gauntlet here.