The production cross sections of ({\text{B}}{\text{s}}^{0}) mesons and charge conjugates are measured in proton-proton (pp) and PbPb collisions via the exclusive decay channel ({\text{B}}{\text{s}}^{0}\to \text{J}/\psi \varphi \to {\mu }^{+}{\mu }^{-}{\text{K}}^{+}{\text{K}}^{-}) at a center-of-mass energy of 5.02 TeV per nucleon pair and within the rapidity range (|y|<2.4) using the CMS detector at the LHC. The pp measurement is performed as a function of transverse momentum (({p}{\text{T}})) of the ({\text{B}}{\text{s}}^{0}) mesons in the range of 7 to 50 GeV/c and is compared to the predictions of perturbative QCD calculations. The ({\text{B}}{\text{s}}^{0}) production yield in PbPb collisions is measured in two ({p}{\text{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}{\text{AA}})) is found to be (1.5±0.6\left(\text{stat}\right)±0.5\left(\text{syst}\right)) for 7–15 GeV/c, and (0.87±0.30\left(\text{stat}\right)±0.17\left(\text{syst}\right)) for 15–50 GeV/c, respectively. Within current uncertainties, the ({\text{B}}{\text{s}}^{0}) results are consistent with models of strangeness enhancement, and suppression by parton energy loss, as observed for the ({\text{B}}^{+}) mesons.