As my current levequest project is 78% done and I just did a quick search of all levequests from Unsynwilf Greensleeves, I can say with 99.5% certainty popcorn does not show up in any CUL leve or quest.
That said, millioncorn has been grown in Eorzea since the Year 1500, they have the means to make "popcorn" and several CUL levequests and recipes do make references to other millioncorn-made foods, like grits. It wouldn't be unreasonable to have popcorn already invented or to just have a restaurant invent it themselves. The means and technology are there, even if the only people to have hypothetically thus far made popcorn are from Mamook in the west.
That said, millioncorn has been grown in Eorzea since the Year 1500, they have the means to make "popcorn" and several CUL levequests and recipes do make references to other millioncorn-made foods, like grits. It wouldn't be unreasonable to have popcorn already invented or to just have a restaurant invent it themselves. The means and technology are there, even if the only people to have hypothetically thus far made popcorn are from Mamook in the west.