We examine possible correlations between $b \to s \ell_1^- \ell_2^+$ transitions $-$ both in the lepton flavor conserving ($\ell_1=\ell_2$) and violating case ($\ell_1 \neq \ell_2$) $-$ and purely leptonic flavor violating decays within the ABCD model, a minimal abelian extension of the Standard Model (SM) introducing a new $\text{U}(1)'$ symmetry. The associated neutral $Z'$ boson has generation-dependent, flavor non-universal couplings to SM fermions, governed by three rational parameters $\epsilon_{1,2,3}$, which sum to zero to ensure gauge anomaly cancellation. Each $\epsilon_i$ is common to all fermions of a given generation, thus inducing correlations among quark and lepton observables. For lepton flavor conserving (LFC) processes, only small deviations from SM predictions were found, reflecting the mutual constraints between the quark and lepton sectors, which preclude large discrepancies. On the other hand, the model allows tree-level lepton flavor violating (LFV) decays, yielding correlations between LFV $b\to s$ transitions and charged lepton decays. The analysis of such correlations shows that the current experimental upper bounds for the rates of $\tau^- \to \mu^-\mu^+\mu^-$, $ \ \mu^-\to e^- \gamma$, $ \ \mu^- \to e^- e^+e^- \, $ and $ \ \mu^- \to e^-$ conversion in nuclei constrain branching ratios of LFV $B_{(s)}$ decays in hierarchical order.