Make the salsa: Combine the pineapple, red onion, cilantro, lime juice, lime zest, and oil in a medium bowl. Season with salt and pepper to taste. Set aside.
Rub the chicken with 2 teaspoons vegetable oil. Sprinkle with the jerk seasoning and ½ teaspoon salt (if your jerk seasoning is salt-free).
Heat 1 tablespoon oil in a large skillet over medium heat. Add the chicken and cook 5 to 6 minutes on each side or until a meat thermometer inserted in the thickest part registers 165°F. Remove the chicken to a cutting board and let stand 5 minutes before thinly slicing against the grain and placing in a medium bowl.
While the chicken rests, add the pepper and onion mix to the drippings in the skillet. Season with ¼ teaspoon salt and pepper. Cook over medium-high heat until the vegetables are tender. Combine with the sliced chicken in the bowl.
Place 4 of the tortillas on a clean work surface. Tear 4 of the cheese slices into smaller pieces and place on each of the tortillas. Top evenly with the chicken mixture and the remaining cheese. Place the remaining tortillas on top.
Wipe out the skillet. Heat 2 tablespoons oil in the skillet over medium heat. Cook the quesadillas, one at a time, until golden brown on each side about 2 minutes per side. Remove the quesadillas from the skillet and cut into wedges before serving with the salsa.