  We present early results from the Epoch of Giant Planet Migration program, a
precise RV survey of over one hundred intermediate-age ($\sim$20$-$200 Myr) G
and K dwarfs with the Habitable-Zone Planet Finder spectrograph (HPF) at
McDonald Observatory's Hobby-Eberly Telescope (HET). The goals of this program
are to determine the timescale and dominant physical mechanism of giant planet
migration interior to the water ice line of Sun-like stars. Here, we summarize
results from the first 14 months of this program, with a focus on our custom RV
pipeline for HPF, a measurement of the intrinsic near-infrared RV activity of
young Solar analogs, and modeling the underlying population-level distribution
of stellar jitter. We demonstrate on-sky stability at the sub-2 m s$^{-1}$
level for the K2 standard HD 3765 using a least-squares matching method to
extract precise RVs. Based on a subsample of 29 stars with at least three RV
measurements from our program, we find a median RMS level of 34 m s$^{-1}$.
This is nearly a factor of 2 lower than the median RMS level in the optical of
60 m s$^{-1}$ for a comparison sample with similar ages and spectral types as
our targets. The observed near-infrared jitter measurements for this subsample
are well reproduced with a log-normal parent distribution with $\mu=4.15$ and
$\sigma=1.02$. Finally, by compiling RMS values from previous planet search
programs, we show that near-infrared jitter for G and K dwarfs generally decays
with age in a similar fashion to optical wavelengths, albeit with a shallower
slope and lower overall values for ages $\lesssim$1 Gyr.
