Kvitelashvili competed at two Grand Prix events, placing fifth at the 2017 Rostelecom Cup and 6th at the 2017 Internationaux de France. He was invited to the Russian event as a replacement for Keiji Tanaka. He won medals at both of his Challenger Series events, taking silver at the 2017 CS Minsk-Arena Ice Star and gold at the 2017 CS Golden Spin of Zagreb.