[5] Shortly before the beginning of the 2023 season, Kvernadze signed for newly-promoted Liga 2 side Kolkheti Poti, before subsequently joining Dinamo Batumi on loan.
[citation needed] On 3 July 2023, Kvernadze officially joined Serie A side Frosinone on a season-long loan, with an option to buy.
[6][7] He made his official debut for the club on 11 August, coming on as a substitute for Giuseppe Caso in the second half of a 1–0 Coppa Italia win over Pisa.
[8][9] One week later, he also made his Serie A debut, coming on for Caso in the 67th minute of a 3–1 league defeat to Napoli.
He played in all ten games of Georgia's 2025 UEFA Championship qualifying campaign in addition to both play-offs against Croatia, in which his team prevailed on penalties.