I would say not really. While he did have some fellow Georgians in the upper echelons of Soviet power (Lavrentry Beria, the head of the KBG; Sergo Ordzhonikidze, the head of Soviet industry until 1937), they were really the exception: consider the other members of the inner circle were mainly ethnic Russian (Molotov, Voroshilov, Kirov, Khrushchev, despite being born in Ukraine; the exception would be Kaganovich, who was Jewish). However I would argue that both Ordzhonikidze (at first) and Beria were brought to Moscow and favoured in part because of their shared Georgian heritage, with caveats.

Stalin was far from a Georgian nationalist, and really felt the idea of a Soviet identity and not the bourgeois notion of national identity. This was developed over his years working with the Bolsheviks in the underground, and is a stark contrast from his youth, when he was very much aware, proud even, to be Georgian. That said he didn't ignore his heritage, and was known to speak Georgian at times, especially when angry or when discussing something secretive (the language is complex enough that few non-Georgians learn it).

As to brining up Ordzhonikidze and Beria: Sergo Ordzhonikidze was an Old Bolshevik (meaning someone who joined the Party before the 1917 Revolution), and worked with Stalin to occupy the Caucasus (he led the Red Army invasions of Azerbaijan, Armenia, and Georgia in 1920-21). He spent time as the de facto Bolshevik leader of the Caucasus until 1926, when he was "promoted" to oversee Soviet industry. I use quotations here because Ordzhonikidze was quite happy staying in his homeland, and only moved because Stalin effectively forced him to do so. Ordzhonikidze stayed in similar roles until 1937, when he began to push back against the purges in Soviet industry, fighting against accusations of sabotage and "wrecking" (the official Soviet term). Sensing he was about to be purged himself, Ordzhonikidze shot himself in February 1937, and instead became a celebrated Soviet hero.

While working in the Caucasus, Ordzhonikidze began to mentor a young Bolshevik from Abkhazia, Lavrentry Beria. Though the two were initially quite close (Beria's oldest son was named "Sergo"), Beria was eager to move up in the world, and began to scheme against his former mentor. When Ordzhonikidze moved to Moscow, Beria soon took over his former role as First Secretary of the Transcaucasus (de facto leader), and while serving in this role oversaw the arrest of Ordzhonikidze's brother in November 1936 on dubious charges. This was a power move by Beria, the first sign that Ordzhonikidze's time was over, and likely contributed to his suicide a few months later.

In 1938, a year after Ordzhonikidze's death, Beria moved to Moscow to take over as head of the NKVD (secret police), and in effect oversee the Great Purge. He used this position to both get closer to Stalin, and consolidate his own power base, to prepare to take charge of the Soviet Union when Stalin was finally out of the picture. He did manage that for a few months in 1953 after Stalin's death, but couldn't hold on and was arrested and shot by the end of that year.

As to their connection with Stalin, both Ordzhonikidze and Beria were frequent guests of his: Stalin would have lavish dinners, full of Georgian-influenced foods, and watch films late into the night, and both of his Georgian compatriots would stick around. While they would mainly use Russian to the benefit of everyone else, Georgian was known to be used when Stalin was particularly angry, or when Beria wanted to speak in private. But other than that, Stalin didn't really show any favoritism towards other Georgians, and aside from his meals hardly showed any hint of his heritage.

Both Ordzhonikidze and Beria have English-language biographies that are worth reading: In Stalin's Shadow: The Career of "Sergo" Ordzhonikidze by Oleg Khlevniuk (1995), and Beria: Stalin's First Lieutenant by Amy Knight (1993). For Stalin's side of things both Stalin, Volume II: Waiting for Hitler, 1929–1941 by Stephen Kotkin (2017) and Stalin: The Court of the Red Tsar by Simon Segab Montefiore (2003) are worth looking at. I'd also encourage taking a look at On Stalin's Team: The Years of Living Dangerously in Soviet Politics by Sheila Fitzpatrick (2015) and Stalin and His Hangmen: The Tyrant and Those Who Killed for Him by Donald Rayfield (2004) for a more specific look at Stalin's inner circle.