Seasonal changes in methane background levels and methane spikes have been detected in situ a metre above the Martian surface, and larger methane plumes detected via ground-based remote sensing, however their origin have not yet been adequately explained. Proposed methane sources include the UV irradiation of meteoritic-derived organic matter, hydrothermal reactions with olivine, organic breakdown via meteoroid impact, release from gas hydrates, biological production, or the release of methane from fluid inclusions in basalt during aeolian erosion. Here we quantify for the first time the potential importance of aeolian abrasion as a mechanism for releasing trapped methane from within rocks, by coupling estimates of present day surface wind abrasion with the methane contents of a variety of Martian meteorites, analogue terrestrial basalts and analogue terrestrial sedimentary rocks. We demonstrate that the abrasion of basalt under present day Martian rates of aeolian erosion is highly unlikely to produce detectable changes in methane concentrations in the atmosphere. We further show that, although there is a greater potential for methane production from the aeolian abrasion of certain sedimentary rocks, to produce the magnitude of methane concentrations analysed by the Curiosity rover they would have to contain methane in similar concentrations as economic reserved of biogenic/thermogenic deposits on Earth. Therefore we suggest that aeolian abrasion is an unlikely origin of the methane detected in the Martian atmosphere, and that other methane sources are required.