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(2S) 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=68.5 GeV. The data sample corresponds to an integrated luminosity of 21.7±1.4 nb−1. The J/ψ and ψ(2S) 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±46 nb/nucleon. The ratio of J/ψ and D0 cross-sections is evaluated to (1.06±0.02±0.09)%. The ψ(2S) 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.