Preheat grill to high. Brush the cut side of the lobster with oil and season with salt and pepper. Place the lobster on the grill, shell side down, and grill until lightly golden brown and just cooked through.
Remove the meat from the lobster and coarsely chop.
Heat a tablespoon of oil in a small saute pan on the grates of the grill. Add the garlic and cook until lightly toasted.
Place 4 tortillas on a work surface and divide the half of the cheese, lobster, and garlic among the tortillas. Season with salt and pepper, to taste.
Place another tortilla on top and then divide the remaining cheese, lobster, and garlic over top. Place the remaining tortillas on top to make 2-layer quesadillas.
Brush tops of the tortillas with oil and then place the quesadillas on the grill, oiled side down. Grill until golden brown, about 2 to 3 minutes.
Carefully flip the quesadillas over, close the cover of the grill, and grill until the cheese has melted, about 2 to 3 minutes longer.
Remove from grill and cut each quesadilla into quarters.