Combine quinoa, broccoli, and cheese in a medium bowl. Season with salt and pepper to taste.
Divide mixture evenly onto one half of each tortilla; fold top of tortilla onto filling.
Heat one teaspoon of oil in a medium cast iron skillet over medium heat; swirl to coat.
Place 2 quesadillas in skillet and cook until browned, about 3 minutes on each side. Add another teaspoon of oil to skillet and repeat with remaining quesadillas.
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