He became a pivotal cog in the Georgian franchise featuring in all of the Black Lions matches for the 2022 season, including starting in the final against Tel Aviv Heat.
[9] He started again the week after against Scarlets, scoring 13 points off the tee to help lead Black Lion secure their first ever EPCR Challenge Cup win "humiliating" the Llanelli based side.
[10][11] He started in Black Lions third Rugby Europe Super Cup final in 2023, scoring 2 penalties and 3 conversions not missing a kick off the tee.
[20] In 2022 he joined Khvamli Tbilisi in the Didi 10, he featured once for the side coming off the bench in the final round of the season, beating RC Aia Kutaisi 18–17.
Matkava started at fly-half in all of Georgia's matches in the 2024 Rugby Europe Championship including the final where he scored 16 points in a 36–10 win over Portugal.
He made his first international start at full back in the 2024 mid-year rugby union tests against Fiji in Batumi, He scored all of Georgia's points in the 12–21 loss.