This numerical study demonstrates heat transfer and irreversibility or entropy generation of non-Newtonian power-law Al2O3-H2O (aluminum oxide-water) nanofluids in a square enclosure using multiple-relaxation-time lattice Boltzmann method accelerated by graphics processing unit computing. In this investigation, the effective thermal conductivity and viscosity are variables, and they depend on the fluid temperature and rate of strain, respectively. The enclosure’s left and right walls are uniformly heated with different temperatures, and the upper and lower walls are thermally adiabatic. There is no valid study and results on non-Newtonian fluid using multiple-relaxation-time lattice Boltzmann method for this configuration and hence the novelty of the present results have been ensured. This paper has formulated and appropriately validated the Newtonian and non-Newtonian natural convection problem with the available numerical results. This study includes a set of comprehensive simulations, showing the effects of these fluids’ natural convection by varying three key parameters: the Rayleigh number, the volume fraction of nanoparticles, and the power-law index on the streamlines, isotherms, local and average Nusselt number as well as the local and total entropy generation. The results show that increasing the volume fraction of the nanoparticles from 0% to 2%, the average rate of heat transfer and the total entropy generation increase 6.5% and 7.4%, respectively, while the Rayleigh number, Ra = 105 and the power-law index n = 0.6.