In a medium bowl, whisk together the soy sauce, brown sugar, sesame oil, minced garlic, grated ginger, rice vinegar, and chopped green onion until well combined.
Place the salmon fillets in a shallow dish and pour the marinade over them, ensuring they are well coated. Cover and refrigerate for at least 30 minutes, or up to 2 hours for more flavor.
Preheat your grill to medium-high heat. Lightly oil the grill grates to prevent sticking.
Remove the salmon from the marinade and let any excess drip off. Season with salt and pepper to taste.
Place the salmon skin-side down on the grill. Cook for about 4-5 minutes, then carefully flip and grill for another 3-4 minutes, or until the salmon is cooked through and flakes easily with a fork.
Remove the salmon from the grill and sprinkle with sesame seeds before serving.