We present new H- and K-band spectroscopy for the bulge of M31, taken with the LUCI spectrograph at the Large Binocular Telescope (LBT). We studied radial trends of CO absorption features (namely, CO1.58, CO1.60, CO1.64, CO1.66, CO1.68, CO2.30, CO2.32, and CO2.35) in the bulge of M31, out to a galactocentric distance of ∼100″ (∼380 pc). We find that most COs do not exhibit a strong radial gradient, despite the strong metallicity gradient inferred from the optical spectral range, except for CO1.64, showing a steep increase in the center. We compared the observed line strengths to predictions of different state-of-the-art stellar population models, including an updated version of EMILES models, which also uses the extended IRTF spectral library. The observed COs are close to models' predictions, but in some models they turn out to be underestimated. We find that the lack of radial gradients is due to the combination of increasing CO strength with metallicity and C abundance, and decreasing CO strength with IMF slope and O abundance. We speculate that the steep gradient of CO1.64 might be due to Na overabundance. Remarkably, we were able to fit, at the same time, optical indices and all the NIR COs (except for CO1.68), leaving abundance ratios (i.e., [C/Fe], [O/Fe], and [Mg/Fe]) as free-fitting parameters, imposing age and metallicity constraints from the optical spectral range, with no significant contribution from intermediate-age populations (∼1 Gyr-old). For the majority of the bulge, we find [Mg/Fe] ∼ 0.15 dex, [O/Fe] larger than [Mg/Fe] (by ∼0.1 dex), and C abundance consistent with that of Mg. In the central (few arcsec) region, we still find an enhancement of O and Mg, but significantly lower [C/Fe]. We find that the COs' line strengths of the bulge are significantly lower than those of massive galaxies, possibly because of a difference in carbon abundance, as well as, to some extent, total metallicity.