Heat grill to low; lightly oil grates. Cut off tip of each cob. One at a time, stand each ear in a large wide bowl; with a sharp knife, carefully slice downward to release the kernels (you should have about 2 cups). Discard cobs.
Add sliced scallions, lime juice, and oil to bowl. Season with salt and pepper, and toss to combine. Set corn relish aside.
Place tortillas on a work surface. Dividing evenly, sprinkle half the cheese on bottom half of each tortilla. Top with squash, bell pepper, scallion pieces, and remaining cheese; season with salt and pepper. Fold top half of tortillas over filling to close.
Grill quesadillas, turning once, until browned in spots and cheese has melted, 6 to 8 minutes. Cut into wedges; serve immediately with corn relish.
Combining a range of colorful vegetables ensures a wide variety of tastes and a good sampling of nutrients.