Combine salmon, onion, bell pepper, bread crumbs, beaten egg, 3 cloves garlic, soy sauce, 1 teaspoon lemon juice, and salt in a large bowl; mix until evenly blended. Shape salmon mixture into 8 equal patties.
Heat a large nonstick skillet over medium-high heat. Cook salmon patties in the hot skillet until golden brown, 4 to 6 minutes per side.
Whisk together mayonnaise, remaining 2 cloves garlic, remaining 1 teaspoon lemon juice, basil, Dijon mustard, and lemon pepper in a medium bowl until well mixed. Serve mayonnaise mixture with salmon patties.