A search for heavy resonances decaying into a Higgs boson ($H$) and a new particle ($X$) is reported, utilizing 36.1 fb$\mathrm{<mrow\; data-mjx-texclass="ORD">-1</mrow>}$ of proton-proton collision data at $s=$ 13 TeV collected during 2015 and 2016 with the ATLAS detector at the CERN Large Hadron Collider. The particle $X$ is assumed to decay to a pair of light quarks, and the fully hadronic final state $XH\to qq¯\prime bb¯$ is analysed. The search considers the regime of high $XH$ resonance masses, where the $X$ and $H$ bosons are both highly Lorentz-boosted and are each reconstructed using a single jet with large radius parameter. A two-dimensional phase space of $XH$ mass versus $X$ mass is scanned for evidence of a signal, over a range of $XH$ resonance mass values between 1 TeV and 4 TeV, and for $X$ particles with masses from 50 GeV to 1000 GeV. All search results are consistent with the expectations for the background due to Standard Model processes, and 95% CL upper limits are set, as a function of $XH$ and $X$ masses, on the production cross-section of the $XH\to qq¯\prime bb¯$ resonance.