  We present a study of the molecular gas distribution and kinematics in the
cicumnuclear region (radii $\lesssim 2\,$kpc) of the $z\approx0.061$ quasar
I$\,$Zwicky$\,$1 using a collection of available Atacama Large
Millimeter/submillimeter Array (ALMA) observations of the carbon monoxide (CO)
emission. With an angular resolution of $\sim0.36''$ (corresponding to
$\sim\,400\,\rm pc$), the host galaxy sub-structures including the nuclear
molecular gas disk, spiral arms, and a compact bar-like component are resolved.
We analyzed the gas kinematics based on the CO image cube and obtained the
rotation curve and radial distribution of velocity dispersion. The velocity
dispersion is about $30\,\rm km\,s^{-1}$ in the outer CO disk region and rises
up to $\gtrsim 100\,\rm km\,s^{-1}$ at radius $\lesssim 1\,$kpc, suggesting
that the central region of disk is dynamically hot. We constrain the CO-to-$\rm
H_2$ conversion factor, $\alpha_{\rm CO}$, by modeling the cold gas disk
dynamics. We find that, with prior knowledge about the stellar and dark matter
components, the $\alpha_{\rm CO}$ value in the circumnuclear region of this
quasar host galaxy is $1.55_{-0.49}^{+0.47}\,M_\odot\,\left(\rm
K\,km\,s^{-1}\,pc^2\right)^{-1}$, which is between the value reported in
ultra-luminous infrared galaxies and in the Milky-Way. The central 1$\,$kpc
region of this quasar host galaxy has significant star formation activity,
which can be identified as a nuclear starburst. We further investigate the high
velocity dispersion in the central region. We find that the ISM turbulent
pressure derived from the gas velocity dispersion is in equilibrium with the
weight of the ISM. This argues against extra power from AGN feedback that
significantly affects the kinematics of the cold molecular gas.
