To address discrepancies between bottom-up and top-down inventories of methane emissions, the detection and quantification of methane point source emissions is of critical concern. Multiple airborne imaging spectrometer campaigns have identified the heavy-tailed distribution of point source methane emissions. The quantification of point source plumes is a two-part problem requiring the detection and delineation of plumes, followed by an accurate accounting of their total methane enhancement. Plume detection and delineation has often relied on manual or statistical methods, but automated methods taking into account plume morphology are essential as the volume of imaging spectrometer data rapidly increases. Fully convolutional neural networks (FCNNs) represent a robust solution to this problem allowing for the detection and delineation of methane point source emissions with minimal analyst input. This work demonstrates the applicability of FCNNs for accurate quantification of methane point source emissions by training a model on data from a 2019 Permian Basin survey by the Next Generation Airborne Visible InfraRed Imaging Spectrometer (AVIRIS-NG). FCNNs were trained using plumes that were manually interpreted from matched filter retrievals of methane enhancements. Our methodology was able to accurately detect and delineate methane plumes, and did so with fewer false positives than statistical methods. Given the anticipated satellite imaging spectrometer missions capable of global mapping of point sources, automated deep learning methods will be necessary to deal with methane plume detection in very large volumes of data.