Heat half the oil in a 10-inch nonstick skillet over medium-high heat until shimmering. Add the zucchini, salt, and pepper and cook, stirring occasionally, until the zucchini is browned and tender, 5 to 7 minutes.

Stir the garlic and cumin into the zucchini and cook until fragrant, about 30 seconds. Transfer the mixture to a bowl and wipe the skillet clean with paper towels.

Place the empty skillet over medium heat for 2 minutes to get it hot. Toast one tortilla at a time until soft and puffed slightly at the edges, about 2 minutes per side, then transfer to a cutting board.

Distribute the zucchini mixture, Monterey Jack, queso fresco, and jalapeños over half of each tortilla, leaving a 1/2-inch border at the edges. Fold the tortillas over and press firmly to flatten.

Brush the top of each folded quesadilla generously with the remaining oil and sprinkle with kosher salt. Place the quesadillas in the skillet, oiled-side down, and cook over medium heat until crisp and well-browned, about 2 minutes.

Brush the upward-facing sides with oil and salt, then flip the quesadillas carefully. Cook until the second side is crisp and the cheese has fully melted, 1 to 2 minutes longer.

Transfer to a cutting board and let rest for at least 3 minutes so the filling sets. Slice into wedges and serve while the cheese is still melty.