Yes and no. The word "barbarian" is a tricky thing when talking about ancient Greece. In English, it carries a lot of context derived from Roman encounters with Germanic groups during the Migration Period mixed with centuries of continuing use and additional context when it was applied to other groups. As we use it today, it means uncultured, uncivilized, and brutish, but in ancient history, it meant something different.

The word "barbarian" is derrived from a very similar Ancient Greek word: barbaros. Because there is a direct connection, "barbarian" is usually used when translating Ancient Greek, but the cultural context was very different. In Ancient Greece, barbaros just meant "foreigner." Anyone who didn't speak Greek was a barbaros. So, by default, the great empires and civilizations around Greece were considered "barbarians," simply because they were not Greek. It was not a judgement on their cultural achievements, just a description of their languages.

The Ancient Greeks had immense praise for their neighbors. Egyptian religion, art, history, and medicine were well known and regularly praised by Greek authors. Persia's political and military might was still undeniable even after they had been defeated by numerous Greek armies. The scientific achievements of Babylon were acknowledged, though often filtered through a very poor understanding of Babylonian and Assyrian history.

The Greek understanding of Babylonia/Mesopotamia is probably the group that changed the most after Alexander's conquests. Prior the Hellenistic Period, Greek knowledge of the region was extremely limited. The achievements of ancient Babylonian astronomers were attributed to the Iranian (ie Persian) prophet Zoroaster and the Magi priests that followed him. The idea of the Assyrian and Babylonian empires was mostly known through myths and legends about fictional Assyrian kings. It wasn't really until the Hellenistic period that the Greeks even acknowledged a difference between "Assyrian" and "Babylonian."

All of that said, the Ancient Greeks did have a word which they occasionally used the way we would use "barbarian" in modern English. This was agrios, which is usually translated as something like "savage," "fierce," or "wild." It can be applied to animals the same way we would call something a "wild animal," but it was also applied to people. Usually this was a description of behavior or mood, as in

The friendship which occurs between opposites is terrible and fierce and seldom reciprocal amongst men... (Plato, Laws 837b)

or

So where you have a savage, uneducated ruler as despot, if there were some one in the city far better than he, I suppose the despot would be afraid of him." in an argument. (Plato, Gorgias, 510b)

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Occasionally though, it was applied to those who the Greeks saw as particularly "uncivilized," but it was treated as somewhat more extreme than our use of "barbarian." The most noteworthy example I can think of is in Herodotus' description of the steppe peoples that participated in the Scythian army that led Darius the Great on a wild goose chase.

The Man-eaters are the most savage of all men in their way of life; they know no justice and obey no law. They are nomads, wearing a costume like the Scythian, but speaking a language of their own; of all these, they are the only people that eat men.

Of course, Herodotus' description is probably hyperbolic, but it gives a sense for how the Greeks had to perceive someone to give them that agrios, uncivilized label. Greek authors, including Herodotus, wrote many ethnographies of people they considered backwards, or lesser in some way, but there was not really a single word for that until this more extreme end of the "civilization" spectrum.