Pliny the Elder (AD 23-69) in his Natural History IX 4.11: writes as follows*:*

"M. Scaurus, in his ædileship, exhibited at Rome, among other wonderful things, the bones of the monster to which Andromeda was said to have been exposed, and which he had brought from Joppa, a city of Judæa. These bones exceeded forty feet in length, and the ribs were higher than those of the Indian elephant, while the back-bone was a foot and a half10 in thickness."

About the same time Suetonius in his Life of Augustus 72 writes " His own villas, which were modest enough, he decorated not so much with handsome statues and pictures as with terraces, groves, and objects noteworthy for their antiquity and rarity; for example, at Capreae the monstrous bones of huge sea monsters and wild beasts, called the "bones of the giants," and the weapons of the heroes. "

Those could, of course, be whale bones. On the other hand, smaller fossils of maritime life do get a mention from the likes of Herodotus, Strabo, eratosthenes and Xenophanes.

There are those who think that this vase from a the Boston Museum of Fine Arts depicts ancient bones. It's also on the cover of Adrienne Meyer's book, The First Fossil Hunters: Paleontology in Greek and Roman Times.

In addition to what /u/BruceWareAllen wrote, you might find pertinent a few articles in the FAQ section "Biology, evolution, and fossils" .

"Is it possible that dinosaur fossils played some role in the origin of dragon myths?" by /u/Dashukta

"Is there any evidence of cultures finding dinosaur fossils before the 17th century? If so, what were their reactions or beliefs surrounding the fossils?" by [deleted]

"Did Native Americans have any interaction with dinosaur fossils?" by /u/400-Rabbits

This is not to discourage discussion. More questions, data, and debate are welcome.