Kvitelashvili is the first Georgian skater to win an ISU Grand Prix of Figure Skating event (Rostelecom Cup 2021) and a medal at the ISU Grand Prix (silver at Rostelecom Cup 2018). He is also the first Georgian male skater to take a medal at an ISU Championship (bronze at European Championships 2020). He competed for Russia until the 2015/16 season. His father competed in cycling and his mother is a former figure skater. His parents emigrated from Georgia to Russia. Kvitelashvili returned to competition for the ISU World Team Trophy 2025 when Georgia qualified for the first time for the event.