Using grid based representation in classical tensor product by increasing the dimension of system is expensive. Thanks to Quantum simulations, this approach can make advantages compared to the classical systems. This research aims to present a mesh-based qubit-operator framework for solving the time-independent Schrödinger equation. We begin with a formulation, the diagonal operator which encodes the mesh coordinate and generates the potential term and the shift operators which generate the nearest-neighbor couplings associated with the kinetic energy operator. The framework is validated on harmonic and anharmonic benchmark potentials using exact diagonalization, the Variational Quantum Eigensolver, and Variational Quantum Deflation. The accuracy of the method is evaluated through energy errors, energy quotients, and wave-function fidelities. We further extend the approach to a molecular vibronic Hamiltonian for pyrazine, where vibrational modes are encoded on qubit grids and coupled to an electronic-state qubit. Quantum phase estimation is then used to extract spectral weights and construct the absorption spectrum. The results demonstrate that the proposed qubit-grid representation provides a flexible framework for simulating continuous-variable quantum systems, molecular vibronic Hamiltonians, and related quantum dynamics problems on quantum computing platforms.