We present a measurement of the forward-backward asymmetry in the production of B(±) mesons, A(FB)(B(±)), using B(±)→J/ψK(±) decays in 10.4 fb(-1) of pp̄ collisions at sqrt[s]=1.96 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 ̄b antiquark, to be produced in the direction of the proton beam. We extract A(FB)(B(±)) from a maximum likelihood fit to the difference between the numbers of forward- and backward-produced B(±) mesons. We measure an asymmetry consistent with zero: A(FB)(B(±))=[-0.24±0.41 (stat)±0.19 (syst)]%.