This article reports on the inclusive production cross section of several quarkonium states, J/ψ, ψ(2S), ϒ(1S), ϒ(2S), and ϒ(3S), measured with the ALICE detector at the LHC, in pp collisions at √s = 5.02 TeV. The analysis is performed in the dimuon decay channel at forward rapidity (2.5 < y < 4). The integrated cross sections and transverse-momentum (pT) and rapidity (y) differential cross sections for J/ψ, ψ(2S), ϒ(1S), and the ψ(2S)-to-J/ψ cross section ratios are presented. The integrated cross sections, assuming unpolarized quarkonia, are: σJ/ψ (pT < 20 GeV/c) = 5.88 ± 0.03 ± 0.34 μb, σψ(2S) (pT < 12 GeV/c) = 0.87 ± 0.06 ± 0.10 μb, σϒ(1S) (pT < 15 GeV/c) = 45.5 ± 3.9 ± 3.5 nb, σϒ(2S) (pT < 15 GeV/c) = 22.4 ± 3.2 ± 2.7 nb, and σϒ(3S) (pT < 15 GeV/c) = 4.9 ± 2.2 ± 1.0 nb, where the first (second) uncertainty is the statistical (systematic) one. For the first time, the cross sections of the three ϒ states, as well as the ψ(2S) one as a function of pT and y, are measured at √s = 5.02 TeV at forward rapidity. These measurements also significantly extend the J/ψ pT reach and supersede previously published results. A comparison with ALICE measurements in pp collisions at √s = 2.76, 7, 8, and 13 TeV is presented and the energy dependence of quarkonium production cross sections is discussed. Finally, the results are compared with the predictions from several production models.