Calcite calcines to quicklime at 850°, which is red-orange. Mere campfires commonly reach yellow heat, which is over 1000°, and have often been used to cook shellfish, whose shells are made of calcite and can calcine to quicklime. So I suspect that the discovery of quicklime probably happened numerous times in mere campfires during the first three million years of human history called the Paleolithic. (There might have even been several independent nutcases who decided to experiment with the stuff even after they saw what it did to water.) Lime kilns, though not the same thing as campfires, were commonly wood-fired, even into the 20th century.

In the Neolithic, the humans were firing pottery, which often requires temperatures over 850°, depending on the clay body you're using. It isn't clear whether lime kilns predate the Neolithic, but if they don't, pottery kilns would have occasionally produced quicklime by accident until you figured out that you can't temper your pots with shellfish (or calcite-containing sand) if you're going to fire them over about 800°, which is not particularly difficult. (Modern earthenware is usually fired over 1000°.) That would have made the discovery of quicklime unavoidable, if it didn't happen earlier, and of course potters must spend a great deal of time investigating the properties of their materials and how they respond to firing.