Preheat grill to medium-high (400°F to 450°F) or a grill pan to high. Brush chayote squash and onion with 1 Tbs. canola oil, then sprinkle with salt. Place on preheated grill. Cook, turning once, until charred and tender, 4 to 5 minutes per side for squash (for grill pan, 6 to 7) and 4 minutes per side for onion (for grill pan, 5 to 6). Remove from grill and coarsely chop. Place in a medium bowl and stir in spinach. Set aside.
Sprinkle 1/3 cup cheese on half of each tortilla. Then top evenly with vegetables. Fold tortillas over, and press gently.
Heat 1/2 Tbs. oil in a large nonstick skillet over medium-high. Add 2 quesadillas and cook 1 to 2 minutes until bottom is golden. Flip and cook about 1 minute more. Remove from skillet and cut into wedges. Repeat with remaining 1/2 Tbs. oil and 2 quesadillas. Serve with pico de gallo.
When chopping chayote, cut around small center core and then discard core pieces.