Clean the grates using a wire brush to remove any residue. Preheat the grill to medium-high heat (375-400°F or 190-200°C). If using a charcoal grill, light it and let it burn until covered in white ash. Lightly brush the grates with oil to prevent sticking.
Wash the zucchini and squash, trim the ends, and cut them diagonally into ½-inch slices, aiming for consistency.
In a small bowl, combine olive oil, minced garlic, salt, black pepper, chopped basil, dried oregano, and the juice of half a lemon. Stir well to blend the flavors.
Lay the zucchini and squash slices in a single layer on a baking sheet or large dish. Drizzle the marinade over the slices, ensuring both sides are coated. Allow the veggies to marinate for 15-20 minutes to infuse the flavors.
Place the marinated zucchini and squash slices directly on the preheated grill. Grill for approximately 3-4 minutes per side until tender and grill marks form. Keep a close eye to avoid overcooking. If using a gas grill, close the lid during grilling; if using a charcoal grill, ensure even coal distribution.
Sprinkle-grated Parmesan cheese on top of the grilled zucchini and squash during the last minute of grilling. The cheese will melt and add a delicious finish.
Remove the grilled zucchini and squash from the grill and transfer to a serving platter. Arrange them in an appealing pattern or stack them up for presentation. Serve as a side dish with grilled meats.