Set oven rack about 6 inches from the heat source and preheat the oven’s broiler. Arrange green chile peppers on a baking sheet.
Roast peppers in the preheated oven, turning 1 or 2 times, until skins are charred and blackened, 5 to 10 minutes. Transfer chile peppers to a resealable plastic bag and seal; set aside until steam starts to peel away skins, about 10 minutes. Carefully open bag and peel skins from peppers. Dice peppers.
Dice 1/2 the bell pepper and 2/3 the onion and mix with tomatoes, jalapeno peppers, cilantro, tomato juice, lime juice, garlic, salt, 1/2 teaspoon black pepper, and 1/4 teaspoon garlic salt together in a bowl. Cover pico de gallo and refrigerate.
Heat 1 tablespoon olive oil in a skillet over medium heat. Thinly slice the remaining 1/3 onion and 1/2 bell pepper; cook and stir with chicken and mushrooms in the hot oil until vegetables are just tender, 5 to 10 minutes. Add roasted green chile peppers, chili powder, oregano, 1 pinch garlic salt, and 1 pinch black pepper; cook and stir until fragrant, about 30 seconds. Stir enough enchilada sauce into vegetable mixture just to coat.
Spread pepperjack cheese and Cheddar cheese onto 1/2 of each tortilla. Spread a layer of vegetable mixture over cheese layer. Fold each tortilla over vegetable-cheese layer. Brush the outsides of tortillas with remaining 2 tablespoons olive oil.
Cook quesadillas, working in batches, in a sandwich press or in a skillet over medium heat until browned and cheeses are melted, about 2 minutes; cut into 4 wedges. Serve quesadillas with the pico de gallo.