We consider Gaussian elliptic random matrices
X of a size
\(N \times N\) with parameter
\(\rho \), i.e., matrices whose pairs of entries
\((X_{ij}, X_{ji})\) are mutually independent Gaussian vectors with
\(\mathbb {E}\,X_{ij} = 0\),
\(\mathbb {E}\,X^2_{ij} = 1\) and
\(\mathbb {E}\,X_{ij} X_{ji} = \rho \). We are interested in the asymptotic distribution of eigenvalues of the matrix
\(W =\frac{1}{N^2} X^2 X^{*2}\). We show that this distribution is determined by its moments, and we provide a recurrence relation for these moments. We prove that the (symmetrized) asymptotic distribution is determined by its free cumulants, which are Narayana polynomials of type B:
$$\begin{aligned} c_{2n} = \sum _{k=0}^n {\left( {\begin{array}{c}n\\ k\end{array}}\right) }^2 \rho ^{2k}. \end{aligned}$$