Do the morty cards get used per quest or is it a timed buff?

Both! After you activate a Morty Card, the buff lasts for 24 hours, or 5 quests.

So if you activate a Morty Card, then finish 5 quests, the buff expires. If you activate a Morty Card after you've done all your quests, you'll have the buff for the 5 quests for the next 24 hours.

I hope that makes sense!

That explains it perfectly, thank you!

thank you also