Neutron captures and delayed decays of reaction products are common sources of backgrounds in ultrarare event searches. In this work, we studied 13C(α,n)16O reactions induced by α particles emitted within the calibration sources of the Majorana Demonstrator. These sources are thorium-based calibration standards enclosed in carbon-rich materials. The reaction rate was estimated by using the 6129-keV γ rays emitted from the excited 16O states that are populated when the incoming α particles exceed the reaction Q value. Thanks to the excellent energy performance of the Demonstrator's germanium detectors, these characteristic photons can be clearly observed in the calibration data. Facilitated by Geant4 simulations, a comparison between the observed 6129-keV photon rates and predictions by a talys-based software was performed. The measurements and predictions were found to be consistent, albeit with large statistical uncertainties. This agreement provides support for background projections from (α,n) reactions in future double-beta decay search efforts.