Neutrinoless double-beta decay of nuclei represents one of the most promising methods for uncovering physics beyond the Standard Model. In this context, $^{76}$Ge stands out as a particularly attractive candidate, as it can serve as an intrinsic component in semiconductor detectors. If the neutrinoless process occurs in $^{76}$Ge, its signature would appear as a distinct peak at the $Q$ value of 2039 keV. A neutron activation measurement was performed on a germanium sample isotopically enriched in $^{76}$Ge at the DT neutron generator of TU Dresden. The measurement confirmed the presence of $\gamma$ rays with energies of 2033.1$\pm$0.5 keV, 2035.5$\pm$0.4 keV, and 2040.22$\pm$0.26 keV originating from the decays of $^{74}$Ga and $^{76}$Ga. These $\gamma$ rays lie in close proximity to the expected neutrinoless double-beta decay signal of $^{76}$Ge.