Laptop Cpu Cooling Fan Not Working.....?

Yes, the fan, if plugged into the headers on the Motherboard, should be spinning. Even PWM controlled fans spin unless you manually disable them.
What kind of Laptop do you have? Usually, there's more than 1 fan header in Desktop Motherboard and You can always plug in the fan into a different fan header on the Motherboard to test the fan. but on Laptop motherboard there is only one header