  We study the Laplacian operator $\Delta_{\bar{\partial}}$ associated to a
K\"ahler structure $(\Omega^{(\bullet, \bullet)}, \kappa)$ for the
Heckenberger--Kolb differential calculus of the quantum quadrics
$\mathcal{O}_q(\textbf{Q}_N)$, which is to say, the irreducible quantum flag
manifolds of types $B_n$ and $D_n$. We show that the eigenvalues of
$\Delta_{\bar{\partial}}$ on zero forms tend to infinity and have finite
multiplicity.
