  We use 118 H$\beta$ quasar (QSO) observations in the redshift range $0.0023
\leq z \leq 0.89$ to simultaneously constrain cosmological model parameters and
QSO 2-parameter radius-luminosity ($R-L$) relation parameters in six different
cosmological models. We find that the $R-L$ relation parameters for these QSOs
are independent of the assumed cosmology so these QSOs seem to be
standardizable through the $R-L$ relation (although there is a complication
that might render this untrue). Cosmological constraints obtained using these
QSOs are weak, more favor currently decelerated cosmological expansion, and
typically are in $\sim 2\sigma$ tension with those obtained from a joint
analysis of baryon acoustic oscillation and Hubble parameter measurements.
Extending the $R-L$ relation to a 3-parameter one to try to correct for the
accretion rate effect does not result in a reduction of the cosmological
constraints discrepancy nor does it result in the hoped-for significant
reduction of the intrinsic scatter of the $R-L$ relation.
