1 tablespoon olive oil, plus more for grates
4 ears corn (husks and silks removed), or 1 package (10 ounces) frozen corn kernels, thawed
1 bunch scallions, half of bunch thinly sliced, other half cut into 2-inch lengths
1 tablespoon fresh lime juice
Coarse salt and ground pepper
4 flour tortillas (10-inch)
1 1/2 cups shredded Monterey Jack cheese (6 ounces)
1 yellow squash, halved crosswise and thinly sliced lengthwise
1 orange bell pepper (ribs and seeds removed), thinly sliced
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.