In a small mixing bowl, gently stir together the pineapple, avocado, onion, lime juice, and cilantro; set aside.
In a large mixing bowl, combine the salmon, panko, eggs, green onions, and teriyaki sauce. Form into four 3/4-inch thick patties.
Heat the oil in a large nonstick skillet over medium heat. Add the patties. Cook until golden brown on both sides, 8 to 10 minutes total. Serve with the avocado-pineapple relish.