The phenomenon of buckling in Cu45Zr45Al10 metallic glass nanowires with different slenderness ratios was studied by means of molecular dynamics simulation. The values of critical stress versus slenderness ratio for two nanowire diameters were obtained. We analyzed the results within the framework of the modified Euler theory of buckling, obtaining values for the surface elastic modulus and the residual surface stress for the two different diameters. Our results show that the Cu45Zr45Al10 metallic glass in nanometric size become stiffer and exhibits a lower Young's modulus than that of a bulk sample.