Drizzle shrimp with olive oil in a large bowl. Sprinkle with salt and pepper and toss to coat. Spread on a baking sheet and bake, rotating once, until just cooked through, 7 to 9 minutes.
When shrimp are cool enough to handle, remove and discard peels and tails. Cut shrimp into thirds.
Shred cheese using the large holes on a box grater.
Spread 1 tablespoon mayonnaise on one side of a tortilla. Place tortilla, mayo-side down, in a large skillet. Sprinkle half the cheese evenly over the top of the entire tortilla, then turn heat on to medium-low heat. While pan is heating, sprinkle half of the chopped shrimp, half of the roasted red peppers and half of the cilantro on one half of the tortilla, leaving the other half with just cheese. Let quesadilla cook, checking that the bottom of the tortilla doesn't burn, until cheese is mostly melted, 3 to 5 minutes. Remove skillet from heat and carefully fold the quesadilla in half like a book, folding the cheese side over the toppings. Place quesadilla on a plate and let rest 5 minutes. Repeat with second tortilla and ingredients. Cut into wedges before serving.