Not yet reviewed by Pith; the record is open.

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.