  We derive an expression for the value $\zeta_Q(3)$ of the spectral zeta
function $\zeta_Q(s)$ studied by Ichinose and Wakayama for the non-commutative
harmonic oscillator defined in the work of Parmeggiani and Wakayama using a
Gaussian hypergeometric function. In this study, two sequences of rational
numbers, denoted $J_2(n)$ and $J_3(n)$, which can be regarded as analogues of
the Ap\'ery numbers, naturally arise and play a key role in obtaining the
expressions for the values $\zeta_Q(2)$ and $\zeta_Q(3)$. We also show that the
numbers $J_2(n)$ and $J_3(n)$ have congruence relations like those of the
Ap\'ery numbers.
