A search is presented for a heavy resonance $Y$ decaying into a Standard Model Higgs boson $H$ and a new particle $X$ in a fully hadronic final state. The full Large Hadron Collider run 2 dataset of proton-proton collisions at $\sqrt{s}=13\text{ }\text{ }\mathrm{TeV}$ collected by the ATLAS detector from 2015 to 2018 is used and corresponds to an integrated luminosity of $139\text{ }\text{ }{\mathrm{fb}}^{\ensuremath{-}1}$. The search targets the high $Y$-mass region, where the $H$ and $X$ have a significant Lorentz boost in the laboratory frame. A novel application of anomaly detection is used to define a general signal region, where events are selected solely because of their incompatibility with a learned background-only model. It is constructed using a jet-level tagger for signal-model-independent selection of the boosted $X$ particle, representing the first application of fully unsupervised machine learning to an ATLAS analysis. Two additional signal regions are implemented to target a benchmark $X$ decay into two quarks, covering topologies where the $X$ is reconstructed as either a single large-radius jet or two small-radius jets. The analysis selects Higgs boson decays into $b\overline{b}$, and a dedicated neural-network-based tagger provides sensitivity to the boosted heavy-flavor topology. No significant excess of data over the expected background is observed, and the results are presented as upper limits on the production cross section $\ensuremath{\sigma}(pp\ensuremath{\rightarrow}Y\ensuremath{\rightarrow}XH\ensuremath{\rightarrow}q\overline{q}b\overline{b}$) for signals with ${m}_{Y}$ between 1.5 and 6 TeV and ${m}_{X}$ between 65 and 3000 GeV.