We examine the stellar velocity dispersions ({sigma}) 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 {sigma} 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{mu}m, to obtain {sigma} measurements from the 2.29{mu}m CO band heads ({sigma}CO) and the 0.85{mu}m calcium triplet ({sigma}_CaT_). For the spiral galaxies in the sample, we found that {sigma}_CO_ is smaller than {sigma}_CaT_, with a mean fractional difference of 14.3 per cent. The best fit to the data is given by {sigma}_opt_=(46.0+/-18.1)+(0.85+/-0.12){sigma}_CO_. This '{sigma}-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 {sigma}-discrepancy in dust-poor early-type galaxies, and a much larger discrepancy in dusty merger remnants and ULIRGs. That {sigma}_CO_ is lower than {sigma}opt may also indicate the presence of a dynamically cold stellar population component. This would agree with the spatial correspondence between low-{sigma}_CO_ and young/intermediate-age stellar populations that has been observed in spatially resolved spectroscopy of a handful of galaxies.