Making his ISU Challenger Series (CS) debut, Kvitelashvili placed fifth at the Lombardia Trophy in September 2014. In November, he competed at the 2014 Rostelecom Cup, replacing the injured Kolyada; he finished twelfth at the event, the first senior Grand Prix (GP) assignment of his career. After placing eighth at the 2015 Russian Championships, he was sent to his second Winter Universiade and finished seventh at the competition, held in February 2015 in Granada, Spain.