We have observed a sample of 22 luminous quasars, in the range 2.0<z<2.5, at 1.6 microns with the near-infrared (NIR) spectrograph FSPEC on the Multiple Mirror Telescope. Our sample contains 13 radio-loud and 9 radio-quiet objects. We have measured the systemic redshifts z_(sys) directly from the strong [O III]5007 line emitted from the narrow-line-region. From the same spectra, we have found that the non-resonance broad H$beta$ lines have a systematic mean redward shift of 520+/-80 km/s with respect to systemic. Such a shift was not found in our identical analysis of the low-redshift sample of Boroson & Green. The amplitude of this redshift is comparable to half the expected gravitational redshift and transverse Doppler effects, and is consistent with a correlation between redshift differences and quasar luminosity. From data in the literature, we confirm that the high-ionization rest-frame ultraviolet broad lines are blueshifted ~550-1050 km/s from systemic, and that these velocity shifts systematically increase with ionization potential. Our results allow us to quantify the known bias in estimating the ionizing flux from the inter-galactic-medium J_(IGM) via the Proximity Effect. Using redshift measurements commonly determined from strong broad line species, like Lyalpha or CIV1549, results in an over-estimation of J_(IGM) by factors of ~1.9-2.3. Similarly, corresponding lower limits on the density of baryon Omega_b will be over-estimated by factors of ~1.4-1.5. However, the low-ionization MgII2798 broad line is within ~50 km/s of systemic, and thus would be the line of choice for determining the true redshift of 1.0<z<2.2 quasars without NIR spectroscopy, and z>3.1 objects using NIR spectroscopy.
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