Allow the chicken to come to room temperature while the smoker heats up—about 30 minutes.
Prepare the chicken by patting each quarter dry with a paper towel, then generously season with salt and pepper.
In a small bowl, mix together the butter, garlic, and 1 tablespoon each of thyme and rosemary. Carefully lift the skin off one chicken quarter, and using your fingers, create a pocket between the skin and meat of the thigh and leg. Generously rub some of the butter mixture inside the pocket and over the outside of the skin on each quarter. Set aside.
Pour the apple cider or juice into the drip pan of the smoker, then add the sprigs of rosemary and thyme to the liquid.
Place each chicken quarter directly on the smoker racks, ensuring they don't touch each other. Insert a meat probe into the thickest part of one of the chicken quarters.
If using wood chips* add them to the smoker and replenish every 30 minutes for the first hour. Smoke the chicken for approximately 2 hours or until the probe indicates an internal temperature of 165°F. Once done, remove the chicken from the smoker, cover it with foil, and let it rest for 10 minutes before serving.
Notes
I like to use apple or cherry wood when smoking poultry.