Set the air fryer to heat at 400 F and preheat for 5 minutes. Place grated parmesan, soft butter, garlic, salt, and pepper in a bowl. Mix everything thoroughly until you get a paste.
Rinse a quarter of the turkey and dry thoroughly with paper towels, then spread the butter and garlic paste with a knife. Spread the paste on the other side of the turkey as well.
Transfer a quarter turkey to the basket of the preheated air fryer and roast for 15 minutes on one side. Then turn it to the other side and cook for another 15 minutes, then check the readiness by piercing the meat to the bone with a skewer.
If pinkish juice appears, then most likely the turkey is not yet fully cooked, so you should increase the cycle by 5-10 minutes, turning the turkey several times. If juices are clear and temp is 165 F. it is done.
Serve the finished turkey quarter whole or cut into pieces, with gravy and extra parmesan.