Combine brining ingredients in a large bowl and stir until sugar and salt are dissolved.
Place tuna into an airtight container and pour the brining liquid over the tuna. Cover and place in refrigerator for an hour.
Remove tuna from the brine and rinse under cold water. Pat dry with paper towels.
Combine dry rub ingredients in a small bowl and then liberally sprinkle rub on all sides of the tuna steaks.
Place tuna steaks directly onto the grill grate of a preheated smoker. Smoke the tuna steaks at 225°F until the internal temperature reaches 125°F. This will approximately be 40 to 60 minutes.
Once the tuna's internal temp reaches 125°F, remove them from the smoker and serve.