In short, a little bit during the era of Stalin and Beria, and mostly because of Beria.

First, I'll make the minor distinction that Beria was ethnically Mingrelian, a group closely related to the Georgians but with their own language, who are from the northwest of Georgia (though few outside of Georgia make that distinction, and the Mingrelian language is considered by some to simply be a Georgian dialect. However the difference is more readily made within Georgia). This doesn't make a huge difference, but it is notable as it arguably played a role in what Beria did (see below).

Stalin may have been an ethnic Georgian (rumours that his father had Ossetian heritage have no basis in fact; the truth is no one is quite certain where the Jughashvili name comes from), but he was committed to the Communist idea of internationalism and downplayed the fact he was Georgian, more readily assuming a Soviet identity. Lenin famously accused him of "Great Russian Chauvinism" prior to his death, which is ironic as it meant that Stalin favoured polices that benefited Russia specifically rather than the Soviet peoples as a whole, despite not being Russian. He did not really do anything that overtly benefited Georgia during his time in office, and would only visit during the summers for vacation, and at that he would only stay at his dacha in Abkhazia (an autonomous republic in northwest Georgia, known for its tropical climate and beaches).

Beria however was much more a nationalist. He gained the favour of Stalin while serving as the head of the OGPU (secret police; precursor to NKVD and KGB) of the Transcaucasus Federation (Armenia, Azerbaijan, and Georgia; the three were united into one republic from 1922 until 1936), and was promoted to General Secretary of Georgia during the 1930s, though he soon effectively became the de facto head of Transcaucasia. He used this to move onto the head of the NKVD in 1938, moving to Moscow, where he stayed until his downfall and execution in 1953. In Tbilisi (the capital of Georgia and Transcaucasia) and later Moscow, Beria brought his own associates, promoting them to power. Many of them were ethnic Mingrelians like himself, and this clique was known for their ethnic heritage. When Stalin started to turn against Beria, he launched a purge against Beria’s group, an event known as the “Mingrelian Affair,” and saw many of his people replaced by those loyal to Stalin rather than Beria.

Beria also saw the mass settlement of ethnic Mingrelian and Georgian farmers into Abkhazia. This was ostensibly done to help cultivate land there, but was more to do with eliminating a rival to Beria: Nestor Lakoba, the head of Abkhazia and a close ally of Stalin. The ethnic Abkhaz had a bare majority in the 1920s, and Lakoba enjoyed immense popularity in his region, which was jokingly called “Lakobistan.” As noted Stalin would vacation there yearly, staying with Lakoba, which made Beria uncomfortable with someone else being so intimate with the leader. So he had Lakoba poisoned in 1936 and started his policy, which was effectively ethnic cleansing: thousands of Georgians were brought to Abkhazia in order to disrupt the demographic balance, and it crushed the Abkhaz leadership and brought it firmly under Georgian (read Beria’s) control, though that only led to resentment and further issues down the road (see 1992-93 and 2008 wars in Georgia).

An avid soccer fan, Beria was also not afraid to extend his influence to that realm. As head of the NKVD he was de facto in charge of the Dinamo sports clubs, which were a part of the NKVD apparatus. This included the famous Dinamo Moscow team, and the Georgian Dinamo Tbilisi. Beria exerted influence to ensure top players joined the Moscow and Tbilisi club, and was not afraid to imprison those who defied the “invitation” to either club. Not surprisingly Dinamo Moscow was one of the top clubs in the 1930s and 1940s, while Dinamo Tbilisi soon rose to prominence as well.