Season both sides of chicken breasts with salt and black pepper.
Heat 2 tablespoons olive oil in a skillet over medium heat; cook chicken breasts in the hot oil until they are no longer pink in the center and juices run clear, 5 to 10 minutes per side. An instant-read thermometer inserted into the thickest part of a breast should read at least 165 degrees F (74 degrees C). Cut chicken into 1/2-inch cubes and transfer to a large bowl.
Heat remaining 4 tablespoons olive oil in the same skillet over medium heat; cook and stir zucchini in the hot oil until slightly tender, about 5 minutes. Season with salt and black pepper. Transfer zucchini to a paper towel-lined plate to drain.
Mix zucchini, garbanzo beans, olives, artichoke hearts, and Parmesan cheese with chicken; toss to combine. Chill in refrigerator before serving, about 1 hour.