A direct search for the standard model Higgs boson decaying to a pair of charm quarks is presented. Associated production of the Higgs and $Z$ bosons, in the decay mode $ZH\to {\ell }^{+}{\ell }^{-}c\overline{c}$ is studied. A data set with an integrated luminosity of $36.1{\mathrm{fb}}^{-1}$ of $pp$ collisions at $\sqrt{s}=13\mathrm{TeV}$ recorded by the ATLAS experiment at the LHC is used. The $H\to c\overline{c}$ signature is identified using charm-tagging algorithms. The observed (expected) upper limit on $\sigma (pp\to ZH)\times B(H\to c\overline{c})$ is 2.7 (${3.9}_{-1.1}^{+2.1}$) pb at the 95% confidence level for a Higgs boson mass of 125 GeV, while the standard model value is 26 fb.