The discovery of a luminous radio burst, FRB 200428, with properties similar to those of fast radio bursts (FRB), in coincidence with an X-ray flare from the Galactic magnetar SGR 1935+2154, supports magnetar models for cosmological FRBs. The burst's X-ray to radio fluence ratio, as well as the X-ray spectral shape and peak energy, are consistent with FRB 200428 being the result of an ultra-relativistic shock (powered, e.g., by an ejected plasmoid) propagating into a magnetized baryonrich external medium; the shock simultaneously generates X-ray/gamma-rays via thermal synchrotron emission from electrons heated behind the shock, and coherent radio emission via the synchrotron maser mechanism. Here, we point out that a unique consequence of this baryon-loaded shock scenario is the generation of a coincident burst of high-energy neutrinos, generated by photo-hadronic interaction of relativistic ions−heated or accelerated at the shock−with thermal synchrotron photons. We estimate the properties of these neutrino burst FRB counterparts and find that a fraction ∼ 10 −8 − 10 −5 of the flare energy (or ∼ 10 −4 − 10 −1 of the radio isotropic energy) is channeled into production of neutrinos with typical energies ∼ TeV−PeV. We conclude by discussing prospects for detecting this signal with IceCube and future high-energy neutrino detectors.