Rubuta ci gaba don wannan sakin layi - Yin wasan farko na ISU Challenger Series (CS), Kvitelashvili ya zama na biyar a gasar Lombardia Trophy a watan Satumbar 2014. A watan Nuwamba, ya yi gasa a gasar cin kofin Rostelecom ta 2014, inda ya maye gurbin Kolyada da ya ji rauni; ya gama na goma sha biyu a taron, babban aikin farko na Grand Prix (GP) a cikin aikinsa. Bayan da aka sanya na takwas a gasar zakarun Rasha ta 2015
, an aika shi zuwa Universiade na hunturu na biyu kuma ya gama na bakwai a gasar, wanda aka gudanar a watan Fabrairun 2015 a Granada, Spain.