Heat a large skillet over medium heat and add 1 tablespoon coconut oil.
Add the pineapple, sprinkle with 1/8 teaspoon salt and 1/4 teaspoon cinnamon, and cook, stirring occasionally, until caramelized and golden, about 6–8 minutes; transfer pineapple to a bowl.
Add another 1/2 tablespoon coconut oil to the skillet, then add the sliced onion and julienned red bell pepper with a pinch of salt; sauté until softened, about 3–5 minutes.
Add the shredded chicken to the skillet and cook until heated through, then combine the chicken and sautéed vegetables with the caramelized pineapple in the bowl.
Reduce heat to low and add remaining 1/2 tablespoon coconut oil to the skillet.
Assemble each quesadilla by placing a tortilla in the skillet, sprinkling about 1/4 cup grated cheddar over half the tortilla, adding a portion of the chicken-pineapple mixture and some chopped cilantro, then topping with another 1/4 cup cheese.
Cook until the bottom tortilla is golden, then carefully flip the quesadilla and cook until the other side is golden and the cheese is melted, about 2–3 minutes per side; repeat with remaining tortillas.
Make the strawberry salsa by combining chopped strawberries, chopped tomato, finely diced onion, diced jalapeño, 1/3 cup chopped cilantro, lemon juice, 1/4 teaspoon salt, and 1/4 teaspoon pepper in a bowl; mix well.
Serve the quesadillas hot with the strawberry salsa on the side.