  We give an algebraic derivation of the energy eigenvalues for the
two-dimensional(2D) quantum harmonic oscillator on the sphere and the
hyperbolic plane in the context of the method proposed by Daskaloyannis for the
2D quadratically superintegrable systems. We provide the special form of the
quadratic Poisson algebra for the classical harmonic oscillator system and
deformed it into the quantum associative algebra of the operators. We express
the corresponding Casimir operator for this algebra in terms of the Hamiltonian
and provide the finite-dimensional representations for this quantum associative
algebra by using the deformed parafermionic oscillator technique. The
calculation of the energy eigen-values is then reduced to finding the solution
of the two algebraic equations whose form is universal for all the 2D
quadratically superintegrable systems. The result derived algebraically agrees
with the energy eigenvalues obtained by solving the Schrodinger equation.
