Yes, but it's not universal. To put simply, there are bland, normal gemstones that are more commonly found and used for jewelry and have zero elemental properties whatsover. However, gemstones are used for enchantments in other accessories so it's possible for them to be imbued with elemental properties and such. It's a common process, but does take some work.

That said, there are very special and rare gemstones that naturally have elemental properties to them. But because these are so special and rare, only the highly affluent have managed to get their hands on them. That said, outside of outright theft of such rare and valuable jewels, such gemstones can be found out in the wild. Some research and digging around will give hints of potential places to look for, chiefly locations that have large pockets of magical energy around it. Looking where large clusters of elemental crystals are is a good start. Just be warned, it won't be easy and even after finding possible locations.