Can a surge protector turn on by themselves?

You've answered your own question. Reading your post, it's easy to understand what the problem is. You're (mis)using the surge protector as the main on-off switch for your fan. Although you can keep the surge protector on forever, you should be controlling the fan using the on-off switch on the fan. A fan does not need a surge protector, and it's unclear why you'd use it for that product. Too, a fan is a large inductive device, and that can play havoc with the proper operation of a surge protector.