For the identification of non-trivial quantum phase, we exploit a Bell-type correlation that is applied to the one-dimensional spin-1 XXZ chain. It is found that our generalization of bipartite Bell correlation can take a decomposed form of transverse spin correlation together with high-order terms. The formulation of density-matrix renormalisation group is utilized to obtain the ground state of a given Hamiltonian with non-trivial phase. Subsequently Bell-SLK-type generalized correlation is evaluated through the analysis of the matrix product state. Diverse classes of quantum phase transitions in the spin-1 model are identified precisely through the evaluation of the first and the second moments of the generalized Bell correlations. The role of high-order terms in the criticality has been identified and their physical implications for the quantum phase has been revealed.
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