  Giant planets exhibit diverse orbital properties, hinting at their distinct
formation and dynamic histories. In this paper, using $\textit{Gaia}$ DR3, we
investigate if and how the orbital properties of Jupiters are linked to their
host star properties, particularly their metallicity and age. We obtain
metallicities for main sequence stars of spectral type F, G, and K, hosting
hot, warm, and cold Jupiters with varying eccentricities. We compute the
velocity dispersion of host stars of these three groups using kinematic
information from $\textit{Gaia}$ DR3 and obtain average ages using velocity
dispersion-age relation. We find that host stars of hot Jupiters are relatively
metal-rich ([Fe/H]=$0.18 \pm 0.13$) and young ( median age $3.97 \pm 0.51$ Gyr)
compared to the host stars of cold Jupiters in nearly circular orbits, which
are relatively metal-poor ($0.03 \pm 0.18$) and older (median age $6.07 \pm
0.79$ Gyr). Host stars of cold Jupiters in high eccentric orbits, on the other
hand, show metallicities similar to that of the hosts of hot Jupiters, but are
older, on average (median age $6.25 \pm 0.92$ Gyr). The similarity in
metallicity between hosts of hot Jupiters and hosts of cold Jupiters in high
eccentric orbits supports high eccentricity migration as the potential origin
of hot Jupiters, with the latter serving as the progenitors. However, the
average age difference between them suggests that the older hot Jupiters may
have been engulfed by the star in a timescale of $\sim 6$ Gyr. This allows us
to estimate the value of stellar tidal quality factor $Q'_\ast\sim10^{6\pm1}$.
