The Greeks knew that they were not native to Greece, and the ethnic group native to Greece were called Palasgians. (Edit: this may actually not be true, see Demetres’s answer). There are probably similar examples from other IE groups.

But if they specifically considered Southern Russia their homeland is very unlikely. The migrations took place long before written history and long before navigation and mapmaking were invented. Those that migrated along the coast of the Black Sea to Northern Anatolia might have had a notion that their ancestors lived on the Other Side. But those that migrated through Central Europe and Central Asia probably had only vague ideas where their ancestors came from.

Also, early IE speaking people soon became admixtures of native and immigrant people. So probably many of them thought primarily of the native people as their ancestors even if they spoke an IE language.

Also, consider this: the migrations took place in many stages. If you were a copper age (Corded Ware Culture) inhabitant of Eastern Poland, maybe some of your parents, grandparents and grand-grand-parents were born in what is today Belorus, while you have ancestors in Western Ukraine going back 5–10 generations etc. Going back some twenty-thirty generations maybe you had a few ancestors in Southern Russia. Most of us can’t trace our ancestry that far back even today where we have written records and we know how the continent looks like on satelite photos. But even if they knew everything about those migrations, there is no particular reason they should consider the PIE homeland something special for them. They might as well emphasize even more distant ancestors going back to pre-PIE times where there ancestors may have come from Siberia, Caucasus and other places.