Bring 1/2 cup water and quinoa to a boil in a saucepan over high heat. Reduce the heat to medium-low, cover, and simmer until quinoa is tender and the water has been absorbed, 15 to 20 minutes. Set aside to cool.

Preheat the oven to 350 degrees F (175 degrees C). Line a baking sheet with foil.

Heat olive oil in a skillet over medium heat. Add onion; cook and stir until soft and translucent, about 5 minutes. Add garlic and cook for 1 more minute. Remove from the heat and cool slightly.

Mix brown sugar, 2 teaspoons Worcestershire sauce, and 1 teaspoon water together in a small bowl until a paste forms.

Mix turkey, cooked quinoa, onion mixture, egg, 2 tablespoons Worcestershire sauce, tomato paste, hot sauce, salt, and pepper together in a large bowl until well combined; the mixture will be very moist. Shape into a loaf and place onto the prepared baking sheet. Spread brown sugar paste over the top.

Bake in the preheated oven until no longer pink in the center, about 50 minutes. An instant-read thermometer inserted into the center should read at least 165 degrees F (74 degrees C). Let cool for 10 minutes before slicing and serving.