AbstractThis article reports on the inclusive production cross section of several quarkonium states, $$\textrm{J}/\psi $$ J / ψ , $$\psi \mathrm{(2S)}$$ ψ ( 2 S ) , $$\Upsilon \mathrm (1S)$$ Υ ( 1 S ) , $$\Upsilon \mathrm{(2S)}$$ Υ ( 2 S ) , and $$\Upsilon \mathrm{(3S)}$$ Υ ( 3 S ) , measured with the ALICE detector at the LHC, in pp collisions at $$\sqrt{s} = 5.02$$ s = 5.02  TeV. The analysis is performed in the dimuon decay channel at forward rapidity ($$2.5< y < 4$$ 2.5 < y < 4 ). The integrated cross sections and transverse-momentum ($$p_{\textrm{T}}$$ p T ) and rapidity ($$y$$ y ) differential cross sections for $$\textrm{J}/\psi $$ J / ψ , $$\psi \mathrm{(2S)}$$ ψ ( 2 S ) , $$\Upsilon \mathrm (1S)$$ Υ ( 1 S ) , and the $$\psi \mathrm{(2S)}$$ ψ ( 2 S ) -to-$$\textrm{J}/\psi $$ J / ψ cross section ratios are presented. The integrated cross sections, assuming unpolarized quarkonia, are: $$\sigma _{\textrm{J}/\psi }$$ σ J / ψ  ($$p_{\textrm{T}} <20$$ p T < 20  GeV/c) = 5.88 ± 0.03 ± 0.34$$ ~\mu $$ μ b, $$\sigma _{\psi \mathrm{(2S)}}$$ σ ψ ( 2 S )  ($$p_{\textrm{T}} <12$$ p T < 12  GeV/c) = 0.87 ± 0.06 ± 0.10$$~\mu $$ μ b, $$\sigma _{\Upsilon \mathrm (1S)}$$ σ Υ ( 1 S )  ($$p_{\textrm{T}} <15$$ p T < 15  GeV/c) = 45.5 ± 3.9 ± 3.5 nb, $$\sigma _{\Upsilon \mathrm{(2S)}}$$ σ Υ ( 2 S )  ($$p_{\textrm{T}} <15$$ p T < 15  GeV/c) = 22.4 ± 3.2 ± 2.7 nb, and $$\sigma _{\Upsilon \mathrm{(3S)}}$$ σ Υ ( 3 S )  ($$p_{\textrm{T}} <15$$ 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 $$\Upsilon $$ Υ states, as well as the $$\psi \mathrm{(2S)}$$ ψ ( 2 S ) one as a function of $$p_{\textrm{T}}$$ p T and $$y$$ y , are measured at $$\sqrt{s} = 5.02$$ s = 5.02  TeV at forward rapidity. These measurements also significantly extend the $$\textrm{J}/\psi $$ J / ψ $$p_{\textrm{T}}$$ p T reach and supersede previously published results. A comparison with ALICE measurements in pp collisions at $$\sqrt{s} = 2.76$$ 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.