We used the density functional theory (DFT) based on full-potential linearized augmented plane wave (FP-LAPW) method implemented in WIEN2K code, to predict the structural, magnetic, electronic and elastic properties of the full-Heusler alloys Co2MB (M = V,Mn) and their MB-terminated (001) surfaces behavior. We showed that the studied alloys are more stable in the ferromagnetic order than the non-magnetic order. The calculated spin magnetic moments of Co2VB and Co2MnB were in good agreement with Slater-Pauling rule. The bulk of Co2VB was predicted to be a half-metallic ferromagnet with Eg = 0.63 eV, and the Fermi level lies in the middle of the indirect gap, and the electrons at EF were fully spin-polarized, unlike to Co2MnB which behaves as a metal. The half-metallicity of the bulk Co2VB was destroyed at (001) surface due to the symmetry reduction. The calculated elastic constants showed that compounds are stable and exhibit a ductile behaviour. The Young's modulus 3D representation showed that studied materials display a strong anisotropy.