We present near-infrared (NIR) synthetic spectra based on PHOENIX stellar atmosphere models of typical early and mid-M dwarfs with varied C and O abundances. We apply multiple recently published methods for determining M dwarf metallicity to our models to determine the effects of C and O abundances on metallicity indicators. We find that the pseudo-continuum level is very sensitive to C/O and that all metallicity indicators show a dependence on C and O abundances, especially in lower T _eff models. In some cases, the inferred metallicity ranges over a full order of magnitude (>1 dex) when [C/Fe] and [O/Fe] are varied independently by ±0.2. We also find that [(O-C)/Fe], the difference in O and C abundances, is a better tracer of the pseudo-continuum level than C/O. Models of mid-M dwarfs with [C/Fe], [O/Fe], and [M/H] that are realistic in the context of galactic chemical evolution suggest that variation in [(O-C)/Fe] is the primary physical mechanism behind the M dwarf metallicity tracers investigated here. Empirically calibrated metallicity indicators are still valid for most nearby M dwarfs due to the tight correlation between [(O-C)/Fe] and [Fe/H] evident in spectroscopic surveys of solar neighborhood FGK stars. Variations in C and O abundances also affect the spectral energy distribution of M dwarfs. Allowing [O/Fe] to be a free parameter provides better agreement between the synthetic spectra and observed spectra of metal-rich M dwarfs. We suggest that flux-calibrated, low-resolution, NIR spectra can provide a path toward measuring C and O abundances in M dwarfs and breaking the degeneracy between C/O and [Fe/H] present in M dwarf metallicity indicators.