A search is performed for W$^{\prime}$ bosons decaying to a top and a bottom quark in the all-hadronic final state, in proton-proton collisions at a center-of-mass energy of 13 TeV using data collected by the CMS experiment between 2016 and 2018 and corresponding to an integrated luminosity of 137 fb$^{-1}$. Deep neural network algorithms are used to identify the jet initiated by the bottom quark and the jet containing the decay products of the top quark when the W boson from the top quark decays hadronically. No excess above the estimated standard model background is observed. Upper limits on the production cross sections of W$^{\prime}$ bosons decaying to a top and a bottom quark are set. Both left- and right-handed W$^{\prime}$ bosons with masses below 3.4 TeV are excluded at 95% confidence level and the most stringent limits to date on W$^{\prime}$ bosons decaying to a top and a bottom quark in the all-hadronic final state are obtained.