In this work we consider a 2D Schrödinger initial value problem in which the cubic nonlinearity has been restricted to quasi-resonant interactions, moreover we consider periodic solutions whose ratio between the periods is an algebraic irrational positive number. We analyze the energy transfer for these solutions by estimating how the support in frequency space of an initial datum propagates in time. Moreover we complement the analytic study with numerical experimentation. As a byproduct of our investigation we prove that the quasi-resonant cubic Schrödinger initial value problem we consider, in both the focusing and defocusing case, are globally well-posed for initial data of finite mass. Finally using a newly discovered conservation law we also show that if the initial data of the quasi-resonant initial value problem is compact, then any norm \(H^s, \, s\ge 1,\) of the associate solution is uniformly bounded.