The two degrees of freedom in the usual third harmonic generation effective Hamiltonian can be separated, both classically and quantum mechanically, into a harmonic oscillator and a nonlinear oscillator. In turn, the quantum Hamiltonian of the nonlinear oscillator can be written as a cubic polynomial in the generators of a quasi-exactly solvable Lie algebra, and the physically relevant eigenstates are precisely those that can be determined exactly (although, in general, not explicitly). Since the standard Jeffreys–Wentzel–Kramers–Brillouin methods are not easily applicable to general third-order differential equations, we use a Bohr–Sommerfeld quantization of the classical orbits to obtain approximate explicit formulas for the corresponding quantum eigenvalues.