Yes. Stalin was well known for favouring Georgian wine and mineral water, and would constantly have Georgian food made for him. This was a big enough deal that the Georgian SSR established a trade office in Moscow to ship food and drink up there (not just for Stalin of course, but to the populace as a whole; a very popular restaurant, especially with higher officials, was a Georgian restaurant). He would also talk to Beria, and other ethnic Georgians in the Kremlin, in Georgian at times, often during his late night meetings (Stalin was a known insomniac, and would have guests stay over until the very early hours to sit around and often watch films). Stalin, despite his distaste of nationalism and attempts to forge a "Soviet identity" was still very cognizant of his Georgian roots, and made no effort to hide them.
For some reading on the matter, I'd suggest going through the following:
Knight, Amy. Beria: Stalin's First Lieutenant (1995). The only biography I know of Beria in English, it shows how Beria interacted with Stalin, and how unlike Stalin Beria tried to downplay his heritage (he was Mingrelian, not Georgian; Mingrelians are from Georgia, and closely related to Georgians, enough that outsiders would see them as a branch of Georgians, though within Georgia it is much more complex)
Kotkin, Stephen. Stalin, Volume I: Paradoxes of Power, 1878-1928 (2014). This shows more his development as a Bolshevik and not so much his time in the Kremlin, but explains how he still showed some Georgian identity. Volume II, Stalin: Waiting for Hitler, 1929-1941, is being released at the end of the month, and should get more into the inner workings of the Kremlin.
Montefiore, Simon Sebag. Young Stalin (2008) and Stalin: Court of the Red Tsar (2003). The former is the most comprehensive biography of Stalin's time in Georgia, while the latter gets into a lot of his personal life, and is very well-written.
Reid, Erik. Familiar Strangers: The Georgian Diaspora and the Evolution of Soviet Empire (2016). This covers the impact of Georgian culture on the Soviet Union, and one of my favourite journal articles is incorporated as a chapter inside: Edible Ethnicity How Georgian Cuisine Conquered the Soviet Table, which gets into how Stalin utilised Georgian cuisine at the Kremlin.