A search for standard model Higgs bosons (H) produced with transverse momentum ($p_\mathrm{T}$) greater than 450 GeV and decaying to bottom quark-antiquark pairs ($\mathrm{b\bar{b}}$) is performed using proton-proton collision data collected by the CMS experiment at the LHC at $\sqrt{s} =$ 13 TeV. The data sample corresponds to an integrated luminosity of 137 fb$^{-1}$. The search is inclusive in the Higgs boson production mode. Highly Lorentz-boosted Higgs bosons decaying to $\mathrm{b\bar{b}}$ are reconstructed as single large-radius jets, and are identified using jet substructure and a dedicated b tagging technique based on a deep neural network. The method is validated with Z $\to$ $\mathrm{b\bar{b}}$ decays. For a Higgs boson mass of 125 GeV, an excess of events above the background assuming no Higgs boson production is observed with a local significance of 2.5 standard deviations ($\sigma$), while the expectation is 0.7. The corresponding signal strength and local significance with respect to the standard model expectation are $\mu_\mathrm{H}$ $=$ 3.7 $\pm$ 1.2 (stat) $^{+0.8}_{-0.7}$ (syst) $^{+0.8}_{-0.5}$ (theo) and 1.9 $\sigma$. Additionally, an unfolded differential cross section as a function of Higgs boson $p_\mathrm{T}$ for the gluon fusion production mode is presented, assuming the other production modes occur at the expected rates.