For the marinade, place all ingredients in blender. Cover and blend until smooth.
Make Lemon-Garlic Marinade. Using meat injector, fill injector container to 1-ounce line. Inject marinade into turkey breasts, thighs and legs, every 1 to 2 inches, pushing plunger down slowly. Refill container and continue to inject turkey until marinade is used. (Or pour marinade over turkey in large glass dish.
Sprinkle Cajun seasoning inside cavity and over outside of turkey. Fasten neck skin to back with skewer. Fold wings across back with tips touching. Tuck drumsticks under band of skin at tail. Cover and refrigerate at least 8 hours but no longer than 24 hours.
If using charcoal grill, place drip pan directly under grilling area, and arrange coals around edge of firebox. Heat coals or gas grill for indirect heat.
Insert barbecue meat thermometer in turkey so tip is in thickest part of inside thigh muscle and does not touch bone.
Cover and grill turkey, breast side up, over drip pan or over unheated side of gas grill and 4 to 6 inches from medium heat 3 to 4 hours or until thermometer reads 180°F and juice is no longer pink when center of thigh is cut. Let stand 15 minutes before carving.