The QCD axion is a promising dark matter candidate that may address the strong CP problem and the cosmic matter-antimatter imbalance. We present a search for the axion-photon conversion signal in the magnetosphere of the Galactic Center magnetar PSR J1745-2900. No credible signals are identified, but we obtain broad-band limits on the axion-photon coupling strength over 62% of the axion mass range between 4 and 166 micro-eV (1-40 GHz). The limits rely on a model of the magnetar magnetosphere and on the extrapolated dark matter density at the Galactic Center (which remains poorly constrained). This axion detection technique lacks the sensitivity of laboratory haloscopes but can efficiently survey broad swaths of allowed axion masses.