We consider the effect of the internal nuclear quadrupole interaction on quantum tunneling in complex multi-atomic two-level systems. Two distinct regimes of strong and weak interactions are found. The regimes depend on the relationship between a characteristic energy of the nuclear quadrupole interaction $λ_{*}$ and a bare tunneling coupling strength $Δ_{0}$. When $Δ_{0}>λ_{*}$, the internal interaction is negligible and tunneling remains coherent determined by $Δ_{0}$. When $Δ_{0}<λ_{*}$, coherent tunneling breaks down and an effective tunneling amplitude decreases by an exponentially small overlap factor $η^{*}≪1$ between internal ground states of left and right wells of a tunneling system. This affects thermal and kinetic properties of tunneling systems at low temperatures $T<λ_{*}$. The theory is applied for interpreting the anomalous behavior of the resonant dielectric susceptibility in amorphous solids at low temperatures $T≤ 5$mK where the nuclear quadrupole interaction breaks down coherent tunneling. We suggest the experiments with external magnetic fields to test our predictions and to clarify the internal structure of tunneling systems in amorphous solids. Comment: To appear in the Physical Review B