Heat 1 1/2 tablespoons olive oil in a large sauté pan over medium-high heat until shimmering. Add the sliced mushrooms and the sliced onion. Cook, stirring occasionally, about 8–10 minutes, until the mushrooms release their liquid and the liquid evaporates and the onions are soft and translucent.
Add the minced garlic to the pan, stir, and cook about 30 seconds until fragrant. Remove the pan from the heat and transfer the mushroom-onion mixture to a bowl; set aside.
Wipe the sauté pan clean (or use a second skillet). Add the remaining 1/2 tablespoon olive oil and return the pan to medium heat.
To assemble a quesadilla, lay one whole-wheat tortilla flat in the hot pan. Sprinkle about 1/6 of the grated Monterey Jack cheese over one half of the tortilla, top with about 1/6 of the mushroom mixture, then sprinkle a little more cheese over the mushrooms. Fold the tortilla in half to enclose the filling.
Cook the folded quesadilla 2–3 minutes per side, pressing gently with a spatula, until both sides are golden brown and the cheese is melted. Adjust the heat as needed to avoid burning.
Transfer the cooked quesadilla to a cutting board, let rest 1 minute, then cut into wedges. Repeat steps 4–6 with the remaining tortillas, cheese, and filling.