This paper presents the first measurement of inclusive $J/\ensuremath{\psi}$ production cross section in the forward pseudorapidity region $2.5\ensuremath{\le}|{\ensuremath{\eta}}^{J/\ensuremath{\psi}}|\ensuremath{\le}3.7$ in $p\overline{p}$ collisions at $\sqrt{s}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1.8$ TeV. The results are based on 9.8 ${\mathrm{pb}}^{\ensuremath{-}1}$ of data collected using the D0 detector at the Fermilab Tevatron Collider. The inclusive $J/\ensuremath{\psi}$ cross section for transverse momenta between 1 and 16 GeV/ $c$ is compared with theoretical models of charmonium production.