For this achievement we need to finish Episode 1. The quest we want is
A Final Goodbye
You'll encounter the Chrome Dragon again who is slightly harder this time around. He's weak to Light element attacks but reagrdless he's a pushover as much as the rest of the story bosses.
After his defeat the game will roll credits after the cutscene. You may skip them if you wish.
Now that we are done you'll notice there's still Episode 1 Cutscene Quests still to go. The quest
Is the only one left we need to obtain Quna's partner card. So start and if you so wish immediately skip the scene and you'll have acquired her Partner card. The next quest is the final scene if you want to watch that and head to episode 2.
Now we go pick an Expedition quest (or whatever you want to do now) and board the Gateway ship.
On your left hand side is a window with an i
This is the Ally terminal. Select Quna and invite her to your party.
Now just finish the quest.