Can a surge protector prevent electrocution?
No, a surge protector will not.

An RCD (Residual Current Device) or GFCI (Ground Fault Circuit Interrupter) aka ELCB (Earth Leakage Circuit Breaker) are the devices that can provide protection, BUT you should be aware that nothing is 100% - these devices can fail... 

You should get the cords away from access by your pet, shielding or protective strips or even moving the cables out of reach are possibilities.