Owing to the fact that even today, most of the major fossil discoveries are the result of chance discovery by a farmer or miner, there's little question that ancient people had been unearthing dinosaur fossils through the ages. The belief in dragons could even have been from the frequent discovery of fossils, which is indicated through many early cave drawings. I mean, what would you think if paleolithic or neolithic you dug up pieces of a stegosaurus skeleton?

The closest answer to your question would be Chang Qu who described 'dragon bones', which would have been dinosaur fossils, 2000 years ago in Wucheng, Sichuan, China.

The first scientific description was from the 1676 discovery of a femur bone by Reverend Plot in Sussex England in his "Natural History of Oxfordshire" (although he thought it might be from an elephant or giant man).

Furthermore, Mary Anning reportedly discovered an ichthyosaur with her brother between the years of 1809 to 1811.

However, the first official classification of dinosaurs that we know of was from William Buckland of Oxford, English who discovered and identified the carnivorous Megalosaurus in 1824, followed closely by Gideon Mantell with a few teeth of the Iguanodon (discovered 1819, named in 1825) which led to the first widespread understanding that we are dealing with an extinct group of reptiles.

The rush to search for fossils came after 1858 when William Foulke found the first dinosaur in the United States (Haddonfield, New Jersey) which also happened to be the most complete fossil ever found.

Dr. Adrienne's Mayor's The First Fossil Hunters is a really good go-to for this type of question. It covers possible evidence that many fossils uncovered in the Lower Egypt and circum-Mediterranean environment, and the western half of the Silk Road, suggests that some explorers and historians were aware of ancient "dragons' bones." And of course, the term is used by Chinese alchemists and such.

Note, however, that it wasn't until Sir Richard Owen that the term "dinosaur" had been used, nor did it have any meaning outside the context in which he used it. Fossils discovered centuries before were eventually included into this category, and ancient "antediluvian" beasts like Basilosaurus were recognized as not giant crocodilians but as whales. Folktales and legends suggest that many ancient monsters and mythic creatures could have been inspired by discovery of dinosaurs' bones, but there's been some pushback on some of these claims.

Dr. Witton has attempted to argue that Mayor's suggestion that Protoceratops-like fossils inspired the Griffin, an icon going back a few millennia, might be in error.. (He's preparing this for a larger publication, so I'm citing a blog post for a source.)