Heat olive oil in a skillet over medium-high heat; sauté mushrooms and onion in hot oil until lightly browned, about 1 minute. Add black beans, cilantro, salt, and pepper; continue sautéing until mushrooms and onions are browned, 2 to 3 minutes more. Remove from heat.

Place 1 tortilla in a clean, dry skillet. Spread 1/2 of the mushroom mixture onto 1/2 of tortilla. Sprinkle with mozzarella cheese and lime juice; fold tortilla over filling. Cook over medium heat until quesadilla is browned and crispy, about 2 minutes per side. Transfer quesadilla to a cutting board. Repeat this step with second tortilla.

Cut quesadillas into wedges and serve with salsa.