Place the chicken breasts, taco seasoning, diced green chiles, and salsa into the slow cooker; cook on high for 2–3 hours, or until the internal temperature reaches 165°F (74°C).
Remove the chicken from the slow cooker and shred it with two forks or tongs; return shredded chicken to the cooker juices.
Stir the mayonnaise into the shredded chicken until evenly combined and set the mixture aside.
Heat a large skillet over medium–low heat and place one flour tortilla in the pan to warm slightly.
Sprinkle about 1/4 cup shredded cheese over the tortilla, then top with about 1/2 cup of the chicken mixture, and another 1/4 cup cheese.
Top with a second tortilla and gently press to adhere, then cook 2–3 minutes until the bottom is golden brown.
Press the top tortilla briefly, flip the quesadilla carefully with a spatula, and brown the second side for 2–3 minutes until cheese is melted and both sides are golden.
Remove from the pan, cut each quesadilla into four triangles, and serve hot with shredded lettuce, diced green onion, sour cream, and taco or hot sauce as desired.