Preheat the grill to medium heat. Lightly brush with oil.
Cut the zucchini in half crosswise and then slice lengthwise into ¼-thick pieces.
Place the zucchini into a bowl with the red bell pepper. Toss with the olive oil, salt and pepper until coated.
Grill the zucchini and peppers until tender, about 2 minutes per side for the zucchini and 3 to 4 minutes per side for the peppers. Slice the peppers into strips.
Arrange 2 tablespoons mozzarella on half of each tortilla. Top with the grilled vegetables, avocado and basil. Fold over the tortillas.
Grill the quesadillas until the cheese is melted and the tortillas have grill marks, 1 to 2 minutes per side.