We have proposed in a series of previous articles a method to determine the effective axial-vector current coupling and the strength of the isoscalar proton-neutron pairing interaction for calculating the nuclear matrix elements of the neutrinoless double-$\ensuremath{\beta}$ ($0\ensuremath{\nu}\ensuremath{\beta}\ensuremath{\beta}$) decay by the quasiparticle random-phase approximation (QRPA). The combination of these two parameters has had an uncertainty in the QRPA approach, but now this uncertainty is removed by introducing a mathematical identity derived under the closure approximation to the nuclear matrix element of the $0\ensuremath{\nu}\ensuremath{\beta}\ensuremath{\beta}$ decay. In this article, we apply our method to the $0\ensuremath{\nu}\ensuremath{\beta}\ensuremath{\beta}$ decays of $^{136}\mathrm{Xe}$ and $^{130}\mathrm{Te}$ and show the nuclear matrix elements and reduced half-lives. Our calculation is tested first by a self-check method using the two-neutrino double-$\ensuremath{\beta}$ decay, and this test ensures the application of our method to $^{136}\mathrm{Xe}$. It turns out, however, that our method is not successful in $^{130}\mathrm{Te}$. Further tests are made for our calculation, and satisfactory results are obtained for $^{136}\mathrm{Xe}$.