In a medium bowl, toss shrimp with oil, salt, and pepper until well coated. Thread shrimp onto skewers, place them on grill over medium-hot fire, and cook 3 to 4 minutes per side. To check doneness: Cut into 1 shrimp and look inside to be sure it is opaque all the way through.
Remove shrimp from fire, and, as soon as they are cool enough to handle, slide them off skewers. Chop into bite-sized pieces and put in a large bowl. Add all remaining ingredients except tortillas, mix well, and season to taste with salt and pepper.
Spread a couple spoonfuls of the shrimp-cheese mixture on 8 tortillas. Place remaining 8 tortillas on top of the filling. Place quesadillas on grill over low fire. Cook until golden brown on both sides and cheese has just melted, about 3 minutes. When they have reached this point, remove them from the grill, cut them into quarters, and serve garnished with pickled red onions.