In a small bowl, combine the mayonnaise, lemon juice, capers, and 1/3 of the green onions. Mix well. Keep in the refrigerator until ready to use.
Preheat the grill to high heat.
In another bowl, combine the salmon, bread crumbs, egg, and remaining green onions. Season with salt and pepper. Add more bread crumbs if the mixture is too soft. Shape into patties approximately four inches in diameter.
Adjust the burners to medium heat. Grill the patties for three to four minutes on each side or until the salmon is cooked through.
Spread the caper mayonnaise on the inside of each bun. Place the patty on top and garnish with cucumber slices. Close the burger and enjoy!