Khvicha is the son of former footballer Badri Kvaratskhelia, He has two brothers, and his younger brother Tornike (b. 2010) also plays football. He has been nicknamed “Kvaradona” after the Napoli legend Diego Maradona, with both of them having very similar dribbling styles. At the beginning of his career, in total, Kvaratskhelia made five appearances in all competitions for Dinamo Tbilisi, scoring his first goal in a 1–0 win away to Shukura Kobuleti on 18 November 201. Due to his performances for Georgia’s under-17 team, Kvaratskhelia became the first winner of the newly introduced Aleksandre Chivadze golden medal in 2017, awarded annually by Georgian Football Federation to talented young players. The following year he received a silver medal for his displays for Georgia’s under-19’s. In March 2018, Kvaratskhelia left Dinamo Tbilisi as a result of a contract dispute and subsequently signed for Rustavi on a free transfer. In April 2018, Kvaratskhelia was the subject of interest from German champions Bayern Munich and was in attendance for Bayern’s 0–0 draw against Sevilla in the UEFA Champions League quarter-final. During the 2018 Erovnuli Liga season, Kvaratskhelia scored three goals across 18 appearances for Rustavi. In 2018, The Guardian named Kvaratskhelia among 60 best young players worldwide. Kvaratskhelia playing for Lokomotiv Moscow in 2019. On 15 February 2019, Kvaratskhelia joined Russian Premier League (RPL) club Lokomotiv Moscow on loan. He made his league debut on 10 March as an 86th-minute substitute for Jefferson Farfán against Anzhi Makhachkala. On 1 July, Lokomotiv announced that Kvaratskhelia had left the club after his loan expired. Following the loss of Kvaratskhelia, Lokomotiv Moscow manager Yuri Semin said he was very disappointed after failing to agree on a permanent transfer with Kvaratskhelia as Semin considered him extremely talented.