A search is made for a vector-like $T$ quark decaying into a Higgs boson and a top quark in 13 TeV proton-proton collisions using the ATLAS detector at the Large Hadron Collider with a data sample corresponding to an integrated luminosity of 139 fb$\mathrm{<mrow\; data-mjx-texclass="ORD">-1</mrow>}$. The Higgs-boson and top-quark candidates are identified in the all-hadronic decay mode, where $H\to bb¯$ and $t\to bW\to bqq¯\prime$ are reconstructed as large-radius jets. The candidate Higgs boson, top quark, and associated B-hadrons are identified using tagging algorithms. No significant excess is observed above the background, so limits are set on the production cross-section of a singlet $T$ quark at 95% confidence level, depending on the mass, $mT$, and coupling, $\kappa T$, of the vector-like $T$ quark to Standard Model particles. In the considered mass range between 1.0 and 2.3 TeV, the upper limit on the allowed coupling values increases with $mT$ from a minimum value of 0.35 for 1.07 < $mT$ < 1.4 TeV to 1.6 for $mT$ = 2.3 TeV.