While you can cut directly on some stone countertops, it is not recommended.
Granite is one of the most durable countertop materials you can buy, and it can certainly withstand cuts from a knife blade. However, if your granite has a sealer on it, cutting can eventually wear down the sealant. Over time, this can lead to liquids and food absorbing into the stone, which can cause stains and other issues that require repair.
With unsealed granite, cutting may be be less of an issue, though again not suggested. Generally the stone is hard enough to be cut on without scratching, however it is also hard enough to dull your knife blades rather quickly.
Similarly, quartz is a very durable, scratch resistant countertop that in most cases can be cut on without damaging the stone. Quartz is not completely scratch proof though, so it is still possible to cause some damage.
Other natural stones are softer and are more likely to scratch if a knife is used on the surface. Marble, limestone and sandstone are among these. Many of these stones also require a sealer, which again can be damaged and lead to scratches and stains. Soapstone is a softer stone that scratches quite easily, however it can be restored with mineral oil or by sanding.
While it is possible to cut directly on natural stone, it's generally safer and easier in the long run to use a cutting board.