It's not impossible that people did find fossil remains before paleontology was an actual field of study, but there are certain practical difficulties, as u/MrPaleontologist explains. And before anyone gets onto the "ancient fossils = mythical creatures" train, that's actually modern-day folklore and Not How It Works, as u/itsallfolklore explains.
As always, if anyone else would like to chime in on this topic, please don't let this post stop you! We always welcome new insights on any matter.
I'm going to quote a post I made on this topic a few years ago:
In the region of North America I focus on, dinosaur fossils are exceptionally rare. So I'll hope you'll forgive me for expanding the scope of your question a bit to cover fossils in general.
I'll start with an old example. Indian Knoll is an Archaic site in the west-central Kentucky, dating to between 5000 and 4000 years ago. The most notable features at the site are large shell midden. Freshwater shellfish made up a large portion of the diet at the time, and all those shells had to go somewhere. As the shell mounds built up, they were later re-purposed into funeral mounds for more than a thousand individuals.
Of particular interest here is that fossils also show up in the mound. The most common are brachiopods, mostly from the Ordovician. These closely resemble more familiar shellfish although they belong to a different phylum entirely. They're common in the area, and it seems that the people of Indian Knoll had a habit of collecting them and moving them into the shell mounds. Why they would do this is less clear. Perhaps the shells held some special significance; perhaps some weary parents were just trying to quietly dispose of their kids' inconvenient collection of fossils.
Other fossils found at the site include fragments of Calamites (a tree-sized horsetail from the Carboniferous) and tapir teeth from the Pleistocene. The tapir teeth in particular were found in association with a human burial, and their placement may suggest they were included as part of a medicine bundle or something similar collection of personally significant tokens of ceremonial power.
Jumping forward in time, the first mostly complete skeleton of a mastodon known to Europeans as originally excavated by Native Americans and presented as a gift to the French. These fossils had come from what's now Big Bone Lick State Park, also in Kentucky. The Lenape have a rather dramatic story to explain how the salt lick became filled with the bones of these extinct animals:
The yahquawhee (mastodons) were created to be beasts of burden and to serve humanity, but they rebelled. So great was their size and strength that they sought to dominate all life and ruthlessly attacked all other species. Eventually humans and other animals form an alliance against the yahquawhee and a great battle is fought against them in the Ohio Valley. The Great Spirit descended to earth to observe the outcome. At first, the battle went disastrously for the allies. The yahquawhee were too strong and their hides too thick to be harmed. The blood for all the fallen allies pooled up around the seemingly victorious yahquawhee.
As the earth turned to bloody mud, the yahquawhees' own immense weight proved to be their undoing. Those that didn't drown in their own victory became so mired that they were easily overwhelmed by the surviving allies. Only the leader of the yahquawhee remained. He battled the Great Bear, the strongest warrior on the alliance's side, and won. At this point, the Great Spirit struck the last yahquawhee with a bolt of lightning. He didn't kill the yahquawhee, but did cause him to flee far to the northwest - never to be seen again.
Copying & pasting a relevant chunk from my past comment on Did Native Americans have any interaction with dinosaur fossils?
We have a very clear and recorded account of native peoples interacting with ancient biological remains from both the indigenous records and Spanish accounts in Mesoamerica. These were not dinosaur bones though, which are primarily found in areas of Mexico where not very many people were living at the time of Contact. These bones are instead from Pleistocene fauna, notably the mammoths and mastodons which were endemic to the Valley of Mexico. Amazingly, these creatures were present when the first humans moved into the area, and human depredation in conjunction with climate change is thought to have led to their extinction.
By the late Postclassic (1200-1521 CE) however, there were no accounts of mastodon hunts within living memory. What there were, however, were huge bones in remarkable prevalence throughout the central Mesoamerican region, including the Basin of Mexico, which is where the groups that would eventually be called the Aztecs would make their home.
Those groups, however, were not autochthonous, but instead migrated into the area over a period of centuries. While they adapated to and integrated themselves into the previously existing complex, settled, agricultural societies of the Basin of Mexico, the histories of these people tend to be a bit... vague when describing anything that may have occurred in the Basin and its surrounding valleys prior to their own arrival. Sahagún, for instance, picks up the thread of Aztec history from the Toltecs, before doubling back to describe the migrations of various peoples from Aztlán. Describing the migration of the Mexica and their stop in Teotihuacan, he says that:
they built the pyramid of the sun and the moon very large, just like mountains. It is unbelievable when it is said they are made by hands, but giants still lived there then. Also it is very apparent from the artificial mountains at Cholollan...
The reference to giants that "still lived there" might be better understood via the numerous references to past races of giants. Markman and Markman's (1992) The Flayed God: The Mesoamerican Mythological Tradition shows how variations on the Mesoamerican, and specifically the Nahua, creation myth stated the past existence of giants (quiname in Nahuatl). A prominent variant is from the Historia de los Mexicanos por sus Pinturas showing the first "sun" (i.e., creation) being one where the Earth was inhabited by giants, who peacefully went about eating acorns before being devoured by jaguars to usher in the 2nd Sun. Not all accounts agree on the exact ordering of creation though, and Fernando de Alva Ixtlilxochitl, an Acolhua-Spanish mestizo writer in the late 16th/early 17th centuries places the giants in the second creation, though he notes that later people encountered some survivors of this previous age.
Diego Durán, a Spanish friar who grew up in Mexico in the 16th century, has the most extensive writings on the quiname, stating that they lived to the east of the Basin of Mexico, "where Puebla and Cholula are found." Note that this coincides with Sahagún pointing towards "Cholollan," which is a Nahuatl spelling for Cholula, as the home of giants. Durán gives a dramatic account of the giants, who "led a bestial existence" of hunting, wearing skins, and living in caves, all of which also could apply to stereotypes the settled peoples of Mesoamerica held about the hunter-gatherer Chichimec groups which had migrated into the Basin region so recently. The recent arrivals in Cholula and Tlaxcala coaxed the giants into attending a feast with them, during which the giant's weapons were taken from them. The Cholulans then rushed out from hiding and slaughtered the giants, thus making the land free and safe for people to settle and farm.
There's a huge amount of symbolism here. There is the basic allegory of a nomadic people "killing" their savage side to be free them to embrace civilization. In true Mesoamerican fashion, a "sacrifice" thus giving rebirth. Yet, this transformation is seen as necessary primarily for the people to the east of the Basin (and thus outside the Aztec sphere), who are seen as savages; they had not incorporated the civilized ways of the Toltecs like their (Aztec) brethren in the Basin. The fact that these people would be the adversaries of the Aztecs, and indeed would side with Spanish, is also fraught with meaning. Notably, the attack on the unarmed quiname resembles the surprise attacks the Spanish would use against the Aztecs during Toxcatl and on the Aztec-aligned elites at Cholula.
Perhaps most relevant to the question though, is this passage from Durán:
In some places of that region enormous bones of the giants have been found, which I myself have seen dug up in rugged places many times.
Durán's translator notes, as we've already covered, tales of giants were common in Postclassic Central Mesoamerica, and that bones excavated from Mexico were among some of the first things sent back to Spain, prompting an investigation by the Royal Physician, Francisco Hernandez, who proclaimed them to be from men standing more than five meters tall. Heyden also notes, however, that the bones of Pleistocene megafauna are abundantly found in Mexico, with a museum exhibiting locally-found mammoth bones in the Teotihuacan Valley. There appears to be another, smaller museum for mammoth remains found in the city limits of Mexico City itself.
And that's where the narrative of Aztec and Spanish interactions with ancient remains would end, with tales of giants in the Nahua past easily ascribed to the proliferation of mammoth/mastodon bones found in the region.