We examine the near-infrared (NIR) emission from low-luminosity active galactic nuclei (LLAGNs). Our galaxy sample includes 15 objects with detected 2-10 keV X-ray emission, dynamical black hole mass estimates from the literature, and available Gemini/NIFS integral field spectroscopy data. We find evidence for red continuum components at the center of most galaxies, consistent with the hot dust emission seen in higher-luminosity AGNs. We decompose the spectral data cubes into a stellar and continuum component, assuming the continuum component comes from thermal emission from hot dust. We detect nuclear thermal emission in 14 out of 15 objects. This emission causes weaker CO absorption lines and redder continuum (2.05-2.28 m) in our K-band data, as expected from hot dust around an AGN. The NIR emission is clearly correlated with the 2-10 keV X-ray flux, with a Spearman coefficient of rspearman = 0.69 suggesting a >99% significance of correlation, providing further evidence of an AGN origin. Our sample has typical X-ray and NIR fluxes 3-4 orders of magnitude less luminous than previous work studying the NIR emission from AGNs. We find that the ratio of NIR to X-ray emission increases toward lower Eddington ratios. The NIR emission in our sample is often brighter than the X-ray emission, with our K-band AGN luminosities comparable to or greater than the 2-10 keV X-ray luminosities in all objects with Eddington ratios below 0.01%. The nature of this LLAGN NIR emission remains unclear, with one possibility being an increased contribution from jet emission at these low luminosities. These observations suggest the James Webb Space Telescope will be a useful tool for detecting the lowest-luminosity AGNs.