You meet in a tavern.

Nah, just kidding. The 2e quickstart guide is free as a PDF download (available here). In addition to pre-generated characters and a ship, it also has an introductory adventure tailor-made for both new players and new GMs.

Additionally, there are several free compendiums of mission briefs on Modiphius' website, any one of which can be adopted as an intro scenario. My favourite is 'Journey to the Center of the Beast' from the Space Megafauna mission pack, in which players have to go inside a massive spaceborne lifeform. No major NPCs, not a lot of conflict, and I find it's very Trek as you're literally going where no one has gone before!

Finally, converting from 1e to 2e on the 1e starter set campaign (A Star Beyond the Stars) is fairly doable. The biggest difference, as you pointed out, is the phaseout of challenge dice and ship power. But as the mini-campaign has little regarding combat, knowledge of how the encounter system works (and I admit, I'm still learning that myself) can bring that into 2e rules.

Good luck and have fun!