The measurement of charmonium states produced in proton-neon (pNe) collisions by the LHCb experiment in its fixed-target configuration is presented. The production of J/ψ and ψ(2 S) mesons is studied with a beam of 2.5 TeV protons colliding on gaseous neon targets at rest, corresponding to a nucleon-nucleon centre-of-mass energy sNN=68.5GeV . The data sample corresponds to an integrated luminosity of 21.7 ± 1.4 nb ^{- 1} . The J/ψ and ψ(2 S) hadrons are reconstructed in μ⁺μ^- final states. The J/ψ production cross-section per target nucleon in the centre-of-mass rapidity range y^⋆∈ [- 2.29 , 0] is found to be 506±8±46nb/nucleon . The ratio of J/ψ and D cross-sections is evaluated to (1.06 ± 0.02 ± 0.09) % . The ψ(2 S) to J/ψ relative production rate is found to be (1.67 ± 0.27 ± 0.10) % in good agreement with other measurements involving beam and target nuclei of similar sizes.