Heat 1 1/2 tablespoons olive oil in a large nonstick skillet over medium-high heat. Add the hash browns, sprinkle with 1/4 teaspoon each salt and pepper, and cook until browned, about 12 minutes. Transfer to a bowl.
Heat the remaining 1/2 tablespoon olive oil in the skillet, then add the onion, bell peppers and 1/4 teaspoon salt; cook until golden, about 8 minutes. Add the salmon and cook, stirring occasionally, until just cooked through, about 3 minutes. Return the hash browns to the skillet to heat through, then remove from the heat and stir in 3 tablespoons of the yogurt mixture. Season with salt and pepper.
Stir 1 tablespoon water into the remaining yogurt mixture and toss with the arugula in a medium bowl. Serve the salmon hash with the arugula salad, and lemon wedges, if desired.