[1] After spending a season with Zenit Saint Petersburg, Kvirkvelia joined another Russian club, Rubin Kazan in 2011.

He made his Russian Premier League debut for Rubin Kazan on 22 July 2011 in a game against FC Terek Grozny.

[2] In February 2017, Rubin Kazan and Lokomotiv Moscow agreed a loan deal for Kvirkvelia until the end of the 2016–17 season.

[6] In October 2021, Kvirkvelia moved to the Ukrainian Premier League side Metalist 1925,[7] although his tenure was abruptly ended due to Russian invasion of Ukraine.

[12] On 6 February 2025, Kvirkvelia moved to Torpedo, the runners-up of the previous Erovnuli Liga season, on a two-year deal.