Sorry to hear you are having trouble.
Have you tried closing the Origin and then see if you could complete the quest?
If that does not work could you upload a savegame to a cloud service (DropBox, Google Drive, One Drive, Paste Bin) and post a link here, the developers can take a look at it.
Savegames are located in \Documents\My Games\Star Control\Saves
Thank you for your patience and patronage.