You’re much further ahead of me now so I cannot speak from experience, but I did see one post about it on the game thread - the translation is a bit wonky which I’ve attempted to clean up:Thargoid ship-capture mission is now completed, and I am on the hunt for the Thargoid heavy cruiser, which is proving tedious. Anyone have any clues? I know which galaxy, other that that it's supposed to be at "an outlying planet". I've given it about 5 hours of playing time searching for it so far.
“Started looking for the Cruiser...
I'm unlucky with this mission... Flew from the 48th to the 47th galaxy, found a planet opposing the Thargoids, in hyperspace found a cruiser with security, broke the security (received RoC), captured the cruiser and went to the station (Military, level 19) where I received the task... Everything seems to be fine... I fly to the station, and there "Galcop is looking for a pilot...” etc. They offer to look for Thargoids for 40cr again and no laser for you... Is that how it should be..?”
Sounds to me like you’ve got to find a system under attack and then search for the cruiser in deep space surrounding the system, though the player who made the post seems to have encountered a bug.