While there are many observations of Li in open clusters, there are very few of the companion light element Be. As we have seen in the study of Be in the Hyades by Boesgaard & King, the two elements together provide important and unique information on the extent and nature of interior mixing in solar-like stars. We have obtained high-resolution (45,000) spectra of the Be II resonance lines in the 14 Pleiades and four α Per dwarfs of spectral types F and G with the Keck I telescope and High-Resolution Echelle Spectrometer. The signal-to-noise ratio in the Be spectral region is typically 40 pixel^-1. These two clusters have similar ages and have solar metallicity. Abundances of Be were determined by spectrum synthesis using the newest version of MOOG. For the F dwarfs where there is only a weak Li dip, there is no indication of a Be dip as was found in the Hyades in association with its deep Li dip. Thus, the observed light element depletion in the F dwarfs in the Hyades and in field stars is occurring during main-sequence evolution, and Be depletion does not become evident until ages of more than 100 Myr. The Pleiades G dwarfs are apparently undepleted in Be and the mean value for log N(Be/H) + 12.00 in stars cooler than 6000 K is 1.26 ± 0.10, compared to the Hyades mean of 1.31 ± 0.07. The star-to-star dispersion in Be in the Pleiades is comparable to the quoted errors. The four α Per stars have lower Be abundances than the Pleiades with a mean of 1.02 dex. The differences in these two clusters in their Li and Be abundances relative to the Hyades is thought to be due to their younger age and possibly their lower metallicity.