Preheat the air fryer to 400 degrees F (200 degrees C).

Place puff pastry on a very lightly floured surface and roll to a 9x12-inch rectangle. Cut into twelve 3-inch squares with a sharp knife. Place about 1 teaspoon chocolate-hazelnut spread in the center of each square; sprinkle with hazelnuts. Fold each square diagonally to form a triangle.

Whisk together egg and water in a small bowl. Brush egg wash over the edges of each pastry triangle; press to seal.

Arrange turnovers in a single layer in the fryer basket and cook in batches until well browned, 5 to 7 minutes per batch. Cool turnovers on a rack.