The quantum-number-projected generator coordinate method (QNPGCM) is applied to the neutron-rich Ge and Se isotopes where the monopole and quadrupole pairing plus quadrupole-quadrupole interaction is employed as an effective interaction. The energy spectra calculated with both axial and triaxial deformations are compared to the shell-model results and the experimental data. In both cases, the QNPGCM reproduces well the energy levels of the even-spin yrast states. However, the QNPGCM results by only assuming axial deformations are not satisfactory enough to reproduce the energy levels of the quasi-$\ensuremath{\gamma}$ bands. Taking account of triaxial deformations is essentially important to describe the yrast and quasi-$\ensuremath{\gamma}$ bands simultaneously.