Yes after you've sliced or cut the cucumbers you need to refrigerate them otherwise they can start to spoil and go bad.
A good way to keep sliced or cut cucumbers fresh is to place them in a container with water covering the cucumbers and seal the container.
The water will help keep the cucumbers fresh and prevent them from drying out.
If you plant to eat the sliced or cut cucumbers within a few hours or so then it's okay to leave them out at room temperature but if not you should always refrigerate the sliced cucumbers to prevent spoilage.