A search for the standard model (SM) Higgs boson ($H$) produced with transverse momentum (${p}_{T}$) greater than 450 GeV and decaying to a charm quark-antiquark ($c\overline{c}$) pair is presented. The search is performed using proton-proton collision data collected at $\sqrt{s}=13\text{ }\text{ }\mathrm{TeV}$ by the CMS experiment at the LHC, corresponding to an integrated luminosity of $138\text{ }\text{ }{\mathrm{fb}}^{\ensuremath{-}1}$. Boosted $H\ensuremath{\rightarrow}c\overline{c}$ decay products are reconstructed as a single large-radius jet and identified using a deep neural network charm tagging technique. The method is validated by measuring the $Z\ensuremath{\rightarrow}c\overline{c}$ decay process, which is observed in association with jets at high ${p}_{T}$ for the first time with a signal strength of ${1.00}_{\ensuremath{-}0.14}^{+0.17}(\mathrm{syst})\ifmmode\pm\else\textpm\fi{}\phantom{\rule{0ex}{0ex}}0.08(\mathrm{theo})\ifmmode\pm\else\textpm\fi{}0.06(\mathrm{stat})$, defined as the ratio of the observed process rate to the SM expectation. The observed (expected) upper limit on $\ensuremath{\sigma}(H)\mathcal{B}(H\ensuremath{\rightarrow}c\overline{c})$ is set at 47 (39) times the SM prediction at 95% confidence level.