Combine salmon, onion, bread crumbs, green bell pepper, 3 cloves garlic, egg, soy sauce, 1 teaspoon lemon juice, and salt in a large bowl; mix until evenly blended. Shape salmon mixture into eight patties.
Heat a large non-stick skillet over medium-high heat. Cook salmon patties until golden brown, 4 to 6 minutes per side.
Whisk mayonnaise, 2 cloves garlic, basil, 1 teaspoon lemon juice, Dijon mustard, and lemon pepper in a bowl until well mixed. Serve mayonnaise mixture with salmon patties.