This article reports on the inclusive production cross section of several quarkonium states, J / ψ , ψ ( 2 S ) , Υ ( 1 S ) , Υ ( 2 S ) , and Υ ( 3 S ) , 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 ( p T ) and rapidity ( y ) differential cross sections for J / ψ , ψ ( 2 S ) , Υ ( 1 S ) , and the ψ ( 2 S ) -to- J / ψ cross section ratios are presented. The integrated cross sections, assuming unpolarized quarkonia, are: σ J / ψ  ( p T < 20  GeV/c) = 5.88 ± 0.03 ± 0.34 μ b, σ ψ ( 2 S )  ( p T < 12  GeV/c) = 0.87 ± 0.06 ± 0.10 μ b, σ Υ ( 1 S )  ( p T < 15  GeV/c) = 45.5 ± 3.9 ± 3.5 nb, σ Υ ( 2 S )  ( p T < 15  GeV/c) = 22.4 ± 3.2 ± 2.7 nb, and σ Υ ( 3 S )  ( p T < 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 ( 2 S ) one as a function of p T and y , are measured at s = 5.02  TeV at forward rapidity. These measurements also significantly extend the J / ψ p T 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.