We present a measurement of the forward-backward asymmetry in the production of ${B}^{\ifmmode\pm\else\textpm\fi{}}$ mesons, ${A}_{\mathrm{FB}}({B}^{\ifmmode\pm\else\textpm\fi{}})$, using ${B}^{\ifmmode\pm\else\textpm\fi{}}\ensuremath{\rightarrow}J/\ensuremath{\psi}{K}^{\ifmmode\pm\else\textpm\fi{}}$ decays in $10.4\text{ }\text{ }{\mathrm{fb}}^{\ensuremath{-}1}$ of $p\overline{p}$ collisions at $\sqrt{s}=1.96\text{ }\text{ }\mathrm{TeV}$ collected by the D0 experiment during Run II of the Tevatron collider. A nonzero asymmetry would indicate a preference for a particular flavor, i.e., $b$ quark or $\overline{b}$ antiquark, to be produced in the direction of the proton beam. We extract ${A}_{\mathrm{FB}}({B}^{\ifmmode\pm\else\textpm\fi{}})$ from a maximum likelihood fit to the difference between the numbers of forward- and backward-produced ${B}^{\ifmmode\pm\else\textpm\fi{}}$ mesons. We measure an asymmetry consistent with zero: ${A}_{\mathrm{FB}}({B}^{\ifmmode\pm\else\textpm\fi{}})=\phantom{\rule{0ex}{0ex}}[\ensuremath{-}0.24\ifmmode\pm\else\textpm\fi{}0.41\text{ }\text{ }(\mathrm{stat})\ifmmode\pm\else\textpm\fi{}0.19\text{ }\text{ }(\mathrm{syst})]%$.