The elemental ratio of carbon to oxygen has been identified as a means of comparison between the composition of giant-planet atmospheres and protoplanetary disks. This method relies on accurate quantification of the major carbon compounds, including both volatiles in gaseous form and solid refractory carbon, present in the planet-forming regions of gas-rich disks around young stars. A combination of ground- and space-based infrared observations of the inner regions of protoplanetary disks have revealed several small carbon-bearing molecules such as CO, CO2, C2H2, and HCN in a number of sources. However, another theoretically abundant carbon carrier, CH4, has only been identified in one disk to date. CH4 is a key molecule of interest for observations with the James Webb Space Telescope (JWST) as it may act as both a significant reservoir of volatile carbon and as an indicator of ongoing destruction of refractory carbon materials in disks. We model the physical structure and chemistry of a generic disk around a T Tauri star to estimate the production of CH4 due to the UV photolysis of hydrogenated amorphous carbon on the surface of dust grains. Variations in disk structure and initial composition are explored to characterize the potential range in CH4 production. We use these estimates to make predictions for JWST MIRI Medium Resolution Spectroscopy (MRS) observations of major carbon species in protoplanetary disks under different assumptions of star and disk properties, including various initial elemental abundances.