They sure can. It depends on the model, but the vast majority of hidden cameras (like these) can record on motion detection.
They have advanced recording features that allow you to tell it how and when to record. You can have it record 24/7 on motion detection, you can have it record solid during the day and on motion at night, or you can have it only record at night on motion detection.
Basically, for each hour of the day, you tell it how to record (nonstop, motion, or not at all). This allows you to have it record in just the right way for any application.