The PHENIX experiment at the Relativistic Heavy Ion Collider has measured the differential cross section, mean transverse momentum, mean transverse momentum squared of inclusive J/ψ, and cross section ratio of ψ(2S) to J/ψ at forward rapidity in p+p collisions at s=510 GeV via the dimuon decay channel. Comparison is made to inclusive J/ψ cross sections measured at s=200 GeV and 2.76-13 TeV. The result is also compared to leading-order nonrelativistic QCD calculations coupled to a color-glass-condensate description of the low-x gluons in the proton at low transverse momentum (pT) and to next-to-leading order nonrelativistic QCD calculations for the rest of the pT range. These calculations overestimate the data at low pT. While consistent with the data within uncertainties above ≈3 GeV/c, the calculations are systematically below the data. The total cross section times the branching ratio is BR d 10 GeV/c)=54.3±0.5(stat)±5.5(syst) nb.