Cook onion with remaining ¼ teaspoon salt and ¼ teaspoon pepper in oil in a 10- to 12-inch skillet over moderate heat, stirring occasionally, until golden, about 6 minutes.
Add garlic and cook, stirring, until fragrant, about 1 minute, then transfer to a large bowl.
Put 1 tortilla on a cutting board and spread ½ cup chicken mixture over half of tortilla, then fold other half over to form a half-moon, pressing firmly on seam.
Assemble 7 more quesadillas in same manner.
Heat lightly oiled grill pan over high heat until it begins to smoke, then reduce heat to moderate and grill quesadillas, 2 at a time, turning over once, until cheese is melted and golden brown grill marks appear, about 4 minutes total per batch.
Transfer with a spatula to cutting board and cut in half.