We present a measurement of the forward-backward asymmetry in the production of B+ mesons, A_FB(B+), using B+ -> J/psi K+ decays in 10.4 inverse femtobarns of p p-bar 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-bar 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, using a boosted decision tree to reduce background. Corrections are made for reconstruction asymmetries of the decay products. We measure an asymmetry consistent with zero: A_FB(B+) = [-0.24 +/- 0.41(stat) +/- 0.19(syst)]%. The standard model estimate from next-to-leading-order Monte Carlo is A_FB(B+) = [2.31 +/- 0.34(stat.) +/- 0.51(syst.)]%. There is a difference of approximately 3 standard deviations between this prediction and our result, which suggests that more rigorous determination of the standard model prediction is needed to interpret these results.