In a small bowl, whisk together the teriyaki sauce, sugar, and garlic until the sugar dissolves.
Place the salmon in a large bowl and toss with marinade to coat. Cover and refrigerate for at least 15 minutes.
Toss the cucumbers with the vinegar; cover and refrigerate until ready to serve.
Heat grill to medium. Remove the salmon from the marinade and arrange skin-side down on grill; cover with lid. Grill without turning, 8 minutes.
Using a large spatula, carefully turn salmon and cook, covered, 2 to 3 minutes longer, just until cooked through but still moist and flaky.
Serve salmon with cucumber salad.