Summary: Dynamical symmetries of Hamiltonians quantized models of discrete nonlinear Schrödinger chain (DNLS) and of Ablowitz-Ladik chain (AL) are studied. It is shown that for n-sites the dynamical algebra of DNLS Hamilton operator is given by the \(su(n)\) algebra, while the respective symmetry for the AL case is the quantum algebra \(su_{q}(n)\). The q-deformation of the dynamical symmetry in the AL model is due to the non-canonical oscillator-like structure of the raising and lowering operators at each site. Invariants of motions are found in terms of Casimir central elements of \(su(n)\) and \(su_{q}(n)\) algebra generators, for the DNLS and QAL cases respectively. Utilizing the representation theory of the symmetry algebras we specialize to the n=2 quantum dimer case and formulate the eigenvalue problem of each dimer as a nonlinear (q)-spin model. Analytic investigations of the ensuing three-term nonlinear recurrence relations are carried out and the respective orthonormal and complete eigenvector bases are determined. The quantum manifestation of the classical self-trapping in the QDNLS-dimer and its absence in the QAL-dimer, is analysed by studying the asymptotic attraction and repulsion respectively, of the energy levels versus the strength of nonlinearity. Our treatment predicts for the QDNLS-dimer, a phase-transition like behaviour in the rate of change of the logarithm of eigenenergy differences, for values of the nonlinearity parameter near the classical bifurcation point.