The first measurement of $J/\psi$ and $D^0$ production in PbNe collisions by the LHCb experiment in its fixed-target configuration is reported. The production of $J/\psi$ and $D^0$ mesons is studied with a beam of lead ions with an energy of 2.5 TeV per nucleon colliding on gaseous neon targets at rest, corresponding to a nucleon-nucleon centre-of-mass energy of $\sqrt{s_{\rm NN}}$ = 68.5 GeV. The $J/\psi/D^0$ production cross-section ratio is studied as a function of rapidity, transverse momentum and collision centrality. These data are compared with measurements from $p$Ne collisions at the same energy and show no difference in the observed $J/\psi$ suppresion trend when comparing $p$Ne and PbNe peripheral collisions with PbNe central collisions.