The $\ensuremath{\nu}\mathrm{BDX}\text{\ensuremath{-}}\mathrm{DRIFT}$ Collaboration seeks to detect low-energy nuclear recoils from $\mathrm{CE}\ensuremath{\nu}\mathrm{NS}$ or BSM interactions at FNAL. Backgrounds due to rock neutrons are an important concern. We present a genie and 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 $\ensuremath{\nu}\mathrm{BDX}\text{\ensuremath{-}}\mathrm{DRIFT}$ detector across several beamlines. In the first stage utilizing 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 $\ensuremath{\nu}\mathrm{BDX}\text{\ensuremath{-}}\mathrm{DRIFT}$, 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 $\ensuremath{\nu}\mathrm{BDX}\text{\ensuremath{-}}\mathrm{DRIFT}$ detector allows additional background subtraction. A sample calculation of a $10\text{ }\text{ }{\mathrm{m}}^{3}\ifmmode\cdot\else\textperiodcentered\fi{}\mathrm{yr}$ exposure to the NuMI low-energy beam configuration shows a $\mathrm{CE}\ensuremath{\nu}\mathrm{NS}$ signal-to-noise ratio of $\ensuremath{\sim}2.5$.
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