Why do we often need to consider the square of four momentum transfer $Q$ in scattering experiment or particle physics/cosmology instead of just the $Q$ itself?
The four-momentum of a physical system is a conserved quantity in a particular reference frame. But we need a relativistic invariant theory which doesn't depend on the choice of coordinates. The squared four-momentum is a scalar invariant, i.e., it doesn't depend on any reference frame. Thus a relativistic invariant theory requires a formulation based on invariant quantities.