It's weird that they didn't give Cid Edea's card, it would've made more sense than what they actually did.
And yeah after you beat the CC Queen to trigger the final part of the quest you have to soeak to Dr. Kadowaki and rest, then there's a random chance of the next step triggering in the dorm. I'm using the wiki for this as I didn't remember myself and can't find my old Prima guide.
There's also a separate Queen of Cards quest but that one is way more annoying and fiddly and involves intentionally losing cards to gain others, you might be better off using a guide (or Leon) for that, I never bothered with it at all.
You also have to choose to rest alone and boot the other party members out.
The Queen of Cards sidequest is super annoying. I don't know that I've ever finished it all the way.