Atomistic fracture toughness of pure Magnesium (Mg) and the Magnesium alloy (Mg-Y) was examined using molecular dynamics simulations. The binary interatomic potential between a solute of Yttrium (Y) and the matrix Mg has been developed with two scaling parameters, which describe the strength of interaction and the adjustment of equilibrium position, respectively. The potential can realize the proper defect energy of twin boundary with the solute element in comparison with the ab-initio calculations. The first simulations of pure Mg with four kinds of crack surfaces, (0001),(10-10), (10-11) and, (10-12) suggest that the unstable KI mode fractures happen at the basal (0001) and the primary pyramidal (10-11). The second simulations of c-axis compressive twin boundary with Y-solute conclude that the strong interaction of Y-solute to Mg-matrix change the fracture mode from the unstable crack propagation along the near-field twin boundary to the void nucleation, which expects the increase of toughness.