The magnetic field effects on natural convection of a non-Newtonian power-law nanofluid in the rectangular enclosure have been investigated using the graphics process unit (GPU) accelerated multiple-relaxation-time (MRT) lattice Boltzmann method (LBM). The enclosure is filled up with a power-law non-Newtonian nanofluid with a proper percentage of the nanoparticle volume fraction. The height of the enclosure is twice its width. The left and right walls are heated with constant temperature, and the top and bottom walls are thermally adiabatic. Initially, the code is validated for the Newtonian nanofluid, and then validation is done with non-Newtonian power-law fluids. The numerical results with the effects of magnetic fields are presented in terms of the streamlines, isotherms, temperature distribution, local and average Nusselt number for the shear thinning and thickening nanofluid. The heat transfer rate gets augmented for the shear-thinning fluids $(n<1)$ while that becomes attenuated for the shear-thickening fluids $(n >1).$ Besides, the magnetic field effects reduce the heat transfer rate from the wall to the fluid region.