He was a part of the squad that finished 1st in the Western Group of the 2010–11 Nemzeti Bajnokság II, which promoted the club to the highest tier of Hungarian football league system.

Kvekveskiri spent the first half of the 2011–12 season on loan with second-tier club Szigetszentmiklósi TK, but later returned to Pécsi MFC, making 6 appearances in the top division.

Alashkert, eventually, won the 2016–17 Armenian Premier League with Kvekveskiri making a total of 28 appearances and scoring 1 goal for the club in all competitions during the season.

In June 2017, Kvekveskiri joined Russian club FC Ararat, which based in Moscow and represented the Armenian diaspora in Russia.

[8] During his first season with Fakel, he made 32 appearances in all competitions, scoring 3 goals, including a brace in a 3–3 home draw against FC Krasnodar on 5 November 2022.