Preheat oven to 200°F and line a large baking sheet with foil or parchment. Place beans in a large bowl and use a potato masher or fork to mash them slightly. Add chicken, cheese and jalapeños and stir to combine. Season with salt.
Place 1 tortilla on a work surface and spread a quarter of chicken mixture over half of tortilla. Fold other half of tortilla over to form a half-moon, pressing firmly. Repeat with remaining chicken mixture and tortillas.
Warm a ridged grill pan or a 12-inch skillet over medium heat. Cook 1 quesadilla at a time, turning carefully once, until cheese is melted and tortillas are golden, 2 to 4 minutes per side. Transfer grilled quesadilla to prepared baking sheet and place in oven to keep warm. Repeat with remaining quesadillas. When all quesadillas are grilled, cut each into 4 wedges and serve immediately.