  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 Ly\alpha
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.
