Not really. Though ethnically Georgian, Stalin grew to espouse a more internationalist identity, and saw himself more as a citizen of the USSR and not just a Georgian. Though he did maintain a close group of associates from the Caucasus, including Georgia, and would speak Georgian and eat Georgian food with them, he effectively dropped the Georgian identity.

In terms of governance, after coming to power Stalin left the Caucasus to Beria to rule. Though Stalin would go to Sochi and Abkhazia for vacations, he last visited Tbilisi, the capital of Georgia, in 1935. Beria effectively ruled it as his own fiefdom, notoriously poisoning Nestor Lakoba, the head of Abkhazia and a potential rival to Beria for control in the region, in 1936.

So aside from importing some Georgian wine and introducing Georgian food to the wider USSR (by virtue of eating it and having his associates do so, thus giving it legitimacy), Stalin didn't really care about Georgia after he took over the USSR as a whole.