We examine the stellar velocity dispersions (σ) of a sample of 48 galaxies, 35 of which are spirals, from the Palomar nearby galaxy survey. It is known that for ultra-luminous infrared galaxies (ULIRGs) and merger remnants, the σ derived from the near-infrared CO band heads is smaller than that measured from optical lines, while no discrepancy between these measurements is found for early-type galaxies. No such studies are available for spiral galaxies - the subject of this paper. We used cross-dispersed spectroscopic data obtained with the Gemini Near-Infrared Spectrograph, with spectral coverage from 0.85 to 2.5 μm, to obtain σ measurements from the 2.29 μm CO band heads (σCO) and the 0.85 μm calcium triplet (σCaT). For the spiral galaxies in the sample, we found that σCO is smaller than σCaT, with a mean fractional difference of 14.3 per cent. The best fit to the data is given by σopt = (46.0 ± 18.1) + (0.85 ± 0.12)σCO. This `σ-discrepancy' may be related to the presence of warm dust, as suggested by a slight correlation between the discrepancy and the infrared luminosity. This is consistent with studies that have found no σ-discrepancy in dust-poor early-type galaxies, and a much larger discrepancy in dusty merger remnants and ULIRGs. That σCO is lower than σopt may also indicate the presence of a dynamically cold stellar population component. This would agree with the spatial correspondence between low-σCO and young/intermediate-age stellar populations that has been observed in spatially resolved spectroscopy of a handful of galaxies.