Yes it would be quite apparent to anyone. Stalin was notorious for speaking Russian with a distinctive Georgian accent (depending on the biography your reading, it is mentioned; I'm pretty sure both Montefiore and Kotkin note it). That he was from Georgia wouldn't have been a major issue though, as many of the leading Soviet leaders were ethnically non-Russian: Trotsky was Jewish, Mikoyan was Armenian, Khrushchev and Brezhnev had some Ukrainian ancestry, even Lenin had some mixed heritage (it's never been confirmed, but he likely had some Chuvash or Kalymk ancestry, and possibly even some Jewish ancestors as well). The Soviet Union of course was a multi-ethnic state, and while it did promote Russian culture and language as the leading force within, it did not downplay the non-Russian peoples (and promoted them, at least initially, as I've written about before; see this comment for one example).

As well, Stalin had gained fame in Bolshevik circles for his work in Georgia when he was younger, helping to organize oil workers there and in Baku (now the capital of neighbouring Azerbaijan; it was also part of the Russian Empire and later USSR). While he had stopped using his birth name of Jughashvili (or Dzhugashvili, as the Russian transliteration goes), and he was not one to promote his own ethnic background (this going contrary to socialist ideals of moving past nationalism, which was seen as a plot to keep the proletariat fighting each other, rather than the bourgeois), he also contradictory in this matter: Stalin was fond of eating Georgian food and drink, and kept some Georgians in his closer circle for quite some time (Lavrenti Beria is the most prominent, but I will note here that Beria was actually ethnically Mingrelian, which is a sub-group of Georgians; so while most non-Georgians wouldn't know or care about the difference, it was/is much more pronounced to Georgians), and was known to sometimes speak to Beria and others in Georgian; being a notoriously complex language, no one around them would understand it, so it had the benefit of maintaining secrecy.