Cut onion crosswise into 1/4-inch-thick slices and arrange slices on a tray, keeping them intact. Brush both sides of slices lightly with oil and season with salt and pepper. Grill onion on a lightly oiled rack set 5 to 6 inches over glowing coals 4 minutes on each side, or until lightly charred and softened. Transfer onion as grilled to a bowl, separating rings.
Brush 2 tortillas lightly with oil on one side and put, oiled side down, on a platter. Divide onion, Monterey Jack, and coriander between tortillas and cover with remaining 2 tortillas. Brush tops of quesadillas lightly with oil.
With a metal spatula transfer quesadillas to a rack set 5 to 6 inches over glowing coals and grill until undersides are golden brown, about 1 minute. Sandwiching each quesadillas between 2 metal spatulas, flip quesadillas over and grill until undersides are golden brown, about 1 minute.
Transfer quesadillas to a cutting board and cut into wedges.