Yes!

We have several examples of authors describing fossils. Herodotus describes 'winged serpents' in the Histories (III 107-109), as well as "shells [that] are found upon the mountains of [Egypt]" (II 12). Avicenna discusses the process of fossilization that is taken up until the sixteenth century. Here is where the story gets a little thorny - there are two bifurcating interpretations of ancient paleontology. One is fantastical, ascribing paleontological underpinnings to mythological creatures in a slipshod way; the other reserved and reactionary, pushing against the motif of the ancient paleontologists. Certainly classical authors were aware of fossils, and examined them critically and rigorously. Many authors made intimations toward evolution perhaps informed by paleontology, including Anaximander and Aristotle (and, later, Saint Augustine), but didn’t directly address fossils. The early philosophy of nature is very much worth examining as the foundation for the ancient study of fossils; in the dialogue Theaetetus Plato contends that "wonder is the beginning of philosophy." However, we don’t have significant evidence to support the more remarkable claims of fossils informing mythology.

We also have some scarce examples of material culture incorporating fossils; carved North American Megalodon teeth have been found, Irish ammonites carved as serpents, Native Americans ascribing holiness to mammoth bones. We have anecdotal stories of Moroccan trilobites as the entombed souls of sinners, of Cyclopean mammoths, of serpents across the world. Some of these are probably stretches (and in fact are coopted as arguments against evolution), while others likely have more weight behind them. Certainly we don’t have ancient civilizations reconstructing dinosaur fossils like you’ll find at a natural history museum, though "Romans loved sensational displays of natural curiosities" (Mayor 138; the veracity of the specimen displayed here is questionable); unambiguously classical scholars were aware of wondrous, monstrous bones.

The principal work on the subject is Adrienne Mayor's The First Fossil Hunters, which is a really great, accessible treatment of premodern paleontology. The third chapter directly addresses your question of bones, but the entire book is worth reading. Mayor’s Fossil Legends of the First Americans is a more targeted work looking at the Native American context, and is more anthropological, but both are easy to read and very much worth it.