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For a single particle trapped in a potential, one can discretize the Time Independent Schrodinger Equation and hence find the eigenvalues of the corresponding Hamiltonian by diagonalising numerically.
Is there any similar numerical way of doing this for an $N$-body interacting quantum system, i.e, to find the eigen-energies of the Hamiltonian tensor ($3N$ dimensional, if 3D system with $N$ particles)?
In theory, you can compute the eigen-energies of any Hamiltonian by straightforward diagonalization as a Hamiltonian is simply a matrix in some Hilbert space. This is true for 1-body systems as well as for N-body interacting systems. However, depending on the assumptions you make, your N-body Hamiltonian will probably live in an exponentially large Hilbert space which will make the exact diagonalization almost impossible.
A simple approach would be to use the Gross-Pitaevskii Equation which is a mean-field Schrodinger-like equation that characterizes an N-body system with delta-like interactions between particles. This equation can be solved numerically for a much cheaper cost than the full N-body Schrodinger equation.
However, finding the energy of a quantum system is the core work of a lot of research in physics, so there are many approximate numerical methods to do so. For instance, you can look at Density Functional Theory (DFT), Quantum Monte Carlos (QMC) or Dynamical Mean-Field Theory (DMFT), which are all used in various contexts.
