In this paper we introduce an analytical model for square Gate-All-Around (GAA) MOSFETs including quantum effects. With the model developed, it is possible to provide an analytical description of the 2D inversion charge distribution function (ICDF) in square GAA MOSFETs of different sizes and for all the operational regimes. The accuracy of the model is verified by comparing the data with that obtained by means of a 2D numerical simulator that self-consistently solves the Poisson and Schrödinger equations. The expressions presented here are useful to achieve a good description of the physics of these transistors; in particular, of the quantization effects on the inversion charge. The analytical ICDF obtained is used to calculate important parameters from the device compact modeling viewpoint, such as the inversion charge centroid and the gate-to-channel capacitance, which are modeled for different device geometries and biases. The model presented accurately reproduces the simulation results for the devices under study and for different operational regimes.