The quadrupole moment of the proton drip-line nuclide 13O(Iπ=3/2−,T=3/2,T1/2=8.6 ms) has been determined for the first time by detecting its β-NQR signal following the production of polarized 13O through the projectile fragmentation process in the 16O + Be collision. The quadrupole moment has been determined to be |Q(13O)|=11.0±1.3 mb. The present result together with the quadrupole moment of its mirror partner 13B is compared with the theoretical values obtained by the shell model code, OXBASH, and by the Hartree-Fock calculation with its single proton separation energy taken into account.