The production cross sections of B$^0_\mathrm{s}$ mesons and charge conjugates are measured in proton-proton (pp) and PbPb collisions via the exclusive decay channel B$^0_\mathrm{s}$ $\to$ $\mu^+\mu^-$K$^+$K$^-$ at a centre-of-mass energy of 5.02 TeV per nucleon pair and within the rapidity range $|y|$ $\lt$ 2.4 using the CMS detector at the LHC. The pp measurement is performed as a function of transverse momentum (p$_\mathrm{T}$) of the B$^0_\mathrm{s}$ meson in the range of 7 to 50 GeV/$c$ and is compared to the predictions of perturbative QCD calculations. The B$^0_\mathrm{s}$ production yield in PbPb collisions is measured in two p$_\mathrm{T}$ intervals, 7 to 15 and 15 to 50 GeV/$c$, and compared to the yield in pp collisions in the same kinematic region. The nuclear modification factor ($R_\mathrm{AA}$) is found to be 1.5 $\pm$ 0.6 (stat) $\pm$ 0.5 (syst) for 7-15 GeV/$c$, and 0.87 $\pm$ 0.30 (stat) $\pm$ 0.17 (syst) for 15-50 GeV/$c$, respectively. Within current uncertainties, the results are consistent with models of strangeness enhancement and a suppression as observed for the B$^+$ mesons.