The short answer is no. We know the tuning of the Greek modes (different from our own) and have some idea of the kind of instruments they used but when it comes to what the music sounded like we can really only guess. This is because the few instances of ancient Greek "sheet music" we have found are basically the words to the songs with the desired note written above them. This gives us no indicator of rhythm or any other musical variable like tempo, volume, timbre etc. We could infer that ancient Greek was "flowing" and lacked a metered pace similar to Gregorian chant but this probably isn't an accurate inference as we see in Plato's description of the various qualities of musical modes that there are modes which readied men to go to war (and were probably used with a steady meter to enhance this effect.) There are some pretty good recordings you can get of contemporary musicians replicating what they think ancient Greek music would sound like but it's really just them guessing to the best of their ability.

The closest thing we have is probably Byzantine Chant. The eight tones of the Octoechos are descended from the modes of classical Greek music, but almost certainly with a lot of differences and development over time. What we can say is that the music chanted today in Greek Orthodox monasteries is fairly similar to medieval Greek church music, but how similar that was to earlier forms I can't say. There is at least a direct organic lineage from Classical times, for what that's worth.

Please note: OP is asking if we actually know that Greek music sounded the way we think it sounded, and if so, how. OP is not just looking for recordings or examples of notation.

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