Thorne-Żytkow objects (TŻOs) are a class of stellar object comprised of a neutron star core surrounded by a large and diffuse envelope. Their exterior appearance is identical to red supergiants; the distinctive electromagnetic signature of a TŻO is a suite of unusual chemical abundance patterns, including excesses of Li, Rb, Mo, and Ca. However, electromagnetic observations cannot unambiguously identify the presence of a neutron star core. Detection of continuous gravitational wave emission from a rotating neutron star core would provide strong supporting evidence for the existence of TŻOs. We present a model for gravitational wave detector confirmation of TŻOs and demonstrate that these objects should be detectable with Advanced LIGO. We also investigate possible targets for joint optical and gravitational searches, and comment on prospects for detectability in both current and future gravitational wave detector networks.