Place quinoa and water into a pot and bring to a simmer; lower heat, cover tightly and low simmer for 15 minutes. Turn off heat and leave cover on to steam for 10 more minutes. Place cooked quinoa into a bowl and stir in vinegar, sugar and salt. Chill and set aside until completely cooled.
Chop pickled ginger and set aside. Do not peel the cucumber; julienne slice and set aside. To assemble each sushi roll, place a nori sheet onto a clean, dry sushi rolling mat. Place 3/4 cup of the cooked and seasoned quinoa on the nori sheet and spread evenly over 2/3 of the sheet. Sprinkle 1/2 Tbsp of chopped ginger and then lay 1/4 of the carrots, 1/4 of the cucumbers and 1/4 of the avocado cut into 3 slices onto the middle of the quinoa. Using the mat, roll the sushi firmly and tightly. Wet the edges with water to seal. Repeat with remaining 3 sheets of nori. Cut into 8 slices and serve.