I am not sure what later Greeks understood abut the Hittites, so I will allow someone else with the requisite expertise to approach that, but it is absolutely certain that those who lived in Greece at the same time as the Hittites were in Anatolia certainly were aware of them. Hittite records detail the trade and gift exchange that took place between their own kings and the kings of various neighbouring peoples, in Syria, Cyprus, Crete, and indeed mainland Greece, which they called Ahhiyawa (considered to be a term related to Achaea).

As others have said, there is certainly archeological evidence of trade connections between the Hittite empire and contemporary Greek palace cultures.

But from the wording of the question, I think what OP meant by ‘Classical Greeks’ was the Greeks after the Dark Ages, when the Hittites themselves were long gone. (Could you clarify please, OP?)

The answer is probably no in this case. I am not aware of any Greek attempts to reconstruct the cultural and ethnical history of Anatolia beyond what they could grasp in the relatively recent historical past. I.e. Herodotus, one of the primary sources for Greek perception of the East, would write about the Lydians, the Medes, the Persians; some 200 years prior to his own time, but no more than that. I am also not aware of any discovery of any discovery of Hittite ruins leading to any ancient historian postulating the existence of something like a Hittite culture. I believe it was discovered in the 19th century.