  The $\nu$BDX-DRIFT collaboration seeks to detect low-energy nuclear recoils
from CE$\nu$NS or BSM interactions at FNAL. Backgrounds due to rock neutrons
are an important concern. We present a~\texttt{GENIE} and~\texttt{GEANT4} based
model to estimate backgrounds from rock neutrons produced in neutrino-nucleus
interactions within the rock walls surrounding the underground halls. This
model was bench-marked against the $2009$ COUPP experiment performed in the
MINOS hall in the NuMI neutrino beam, and agreement is found between
experimental results and the modeled result to within $30\%$. Working from this
validated model, a similar two-stage simulation was performed to estimate
recoil backgrounds in the $\nu$BDX-DRIFT detector across several beamlines. In
the first stage utilizing~\texttt{GEANT4}, neutrons were tallied exiting the
walls of a rectangular underground hall utilizing four different neutrino beam
configurations. These results are presented for use by other underground
experiments requiring estimations of their rock neutron backgrounds. For
$\nu$BDX-DRIFT, the second stage propagated neutrons from the walls and
recorded energy deposited within a scintillator veto surrounding the detector
and nuclear recoils within the detector's fiducial volume. The directional
signal from the $\nu$BDX-DRIFT detector allows additional background
subtraction. A sample calculation of a $10\,$m$^3\cdot\,$yr exposure to the
NuMI Low Energy (LE) beam configuration shows a CE$\nu$NS signal-to-noise ratio
of $\sim$2.5.
