Why do we calculate the pitching moment coefficient about quarter chord point, when force and moment system can be transferred to any convenient point?
Because the quarter chord point is the most meaningful point: the wing or surface lift vector can be found here. Like taking the CoG as reference point for gravity forces.
Theoretical and practical analysis have shown that if the moment and force reference point is taken at 25% of Mean Aerodynamic Cord (MAC), the magnitude of the aerodynamic moment remains nearly constant when the angle of attack changes. This location is called the wing's Aerodynamic Centre (AC).
The lift and moment system can be transferred to any convenient point, but that is the point of the quarter chord point :)
The aerodynamic center is the point at which the pitching moment coefficient for the airfoil does not vary with lift coefficient (i.e. angle of attack), so this choice makes analysis simpler
And, the aerodynamic center is found approximately at quarter chord length from the front.