To Cook Broccoli: Bring a large pot of water to a boil. Add one teaspoon of salt. Add broccoli florets and cook until crisp tender, about 2 minutes. Remove, place a bowl with ice water. Drain and chop the broccoli.
Combine broccoli with cheese, corn and garlic powder. Add salt and pepper to taste. Divide mixture in four, for each tortilla. Add the mixture on one half of the tortilla and fold the other half over the mixture.
Grease a large frying pan using 1 teaspoon of oil. Heat frying pan over medium heat, about 3 minutes. Carefully add two folded tortillas side by side and cook until golden. Flip and cook on the other side too. Cheese should be melted and tortillas gold and crisp. Remove from heat. Repeat this step with the other two folded tortillas.
Cut quesadillas in half or in thirds and serve warm.