The New Frontier (Star Trek: Dragon episode)

"The New Frontier" is the Season 1 finale and was written by Jon Wasik.

Overview
While en route to Earth to be debriefed about their encounter with Voyager, the USS Dragon passes through an area of unusual subspace activity. To be safe, they do so at impulse, but once they are clear and try to jump to warp, the field expands and their warp field becomes super charged. The ship accelerates at amazing speed, and in less than a minute, the ship screams through the galactic barrier, accelerating away from the galaxy.

Four hours later, the warp field finally dissipates and they drop out of warp. They learn that they have literally traveled to the other side of the known universe, to the edge of the Kalium Galaxy. Even at maximum warp, it would take tens of thousands of years to reach the Milky Way again. As a result, Captain Harriman orders them further into the Kalium Galaxy to explore and search for advanced technologies to get them home. What they encounter are the Vorkalai, a species that is surprisingly humanoid, but also very violent. One ship attacks the Dragon and very nearly destroys her.

The crew salvages a Vorkalai computer core and learn that the Vorkalai are a species of conquerers, their space expanding towards the Dragon's location. Captain Harriman orders his engineers to use knowledge gained in the other dimensions to modify their warp core, and after a couple of hours, the Dragon achieves transwarp velocities without the help of Borg technology. They set a course away from Vorkalai space with the intent to find a species capable of fighting the Vorkalai to give them advanced warning and to help them fight.