Grill corn, covered, over medium heat for 10 minutes; turn. Place the squash, onion and jalapenos on grill; cover and cook for 5-6 minutes on each side. When vegetables are cool enough to handle, remove corn from the cobs, chop the squash and onion, and seed and chop the jalapenos. Place in a large bowl.
Stir in the basil, oregano, garlic, salt and cumin. Place 1/2 cup filling on 1 side of each tortilla; sprinkle with cheese. Fold tortillas over filling. In a large cast-iron skillet or griddle, cook quesadillas in oil over medium heat until heated through, 1-2 minutes on each side. Cut into wedges.