Let $M_{ij}$ be a real random matrix, constrained to be symmetric $M_{ij}=M_{ji}$, and with zero diagonal, $M_{ii}=0$.
Suppose we know that, for any real vector $v_i$, the following holds:
$$\operatorname{var}\left( \sum_{ij}M_{ij}v_{i}v_{j} \right) = \frac{1}{n} \left[ \left( \sum_{i}v_{i}^{2} \right)^{2} - \sum_{i}v_{i}^{4} \right]$$
where $n$ is the dimensionality, so $i,j\in\{1,\dots,n\}$.
What can we conclude about the variances and covariances of the entries $M_{ij}$?
We start from the equality
$$\operatorname{var}\left( \sum_{ij}M_{ij}v_{i}v_{j} \right) = \frac{1}{n} \left[ \left( \sum_{i}v_{i}^{2} \right)^{2} - \sum_{i}v_{i}^{4} \right]$$
This is an equality of two polynomials in the variables $v_i$. Since the $v_i$ are free, the coefficients of different mixed powers must match.
Let's start with the right-hand side. expanding the variance,
$$\operatorname{var}\left( \sum_{ij}M_{ij}v_{i}v_{j} \right) = \sum_{ijkl} \operatorname{cov}(M_{ij},M_{kl}) v_i v_j v_k v_l$$
For the right-hand side, we have,
$$\left( \sum_{i}v_{i}^{2} \right)^{2} - \sum_{i}v_{i}^{4} = \sum_{ij}v_i^2 v_j^2 - \sum_{i}v_{i}^{4} = \sum_{i \neq j} v_i^2 v_j^2$$
The coefficient of $v_i^2 v_j^2$ on the right is $\frac{2}{n}$ if $i<j$, while terms do not appear (i.e., their coefficient is zero).
From this we conclude that $\operatorname{cov}(M_{ij},M_{kl})=0$ unless the $i,j,k,l$ consists of two repeated distinct indices. The only possibilities are $\operatorname{cov}(M_{ij},M_{ij})=\operatorname{var}(M_{ij})$, and $\operatorname{cov}(M_{ii},M_{jj})=0$ (because $M$ has zero diagonal). Therefore, assuming $i\le j$ and $k\le l$, $$\operatorname{cov}(M_{ij},M_{kl})=\delta_{ik}\delta_{jl}\operatorname{var}(M_{ij})$$
Distinct entries of the matrix $M_{ij}$ must be uncorrelated.
The variances $\operatorname{var}(M_{ij})$ can be computed easily. For instance let $v_i=\delta_{i,1}+\delta_{i,2}$. Then,
$$\operatorname{var}\left( 2M_{12} \right) = \frac{1}{n} \left[ 2^{2} - 2 \right] = \frac{2}{n}$$
Thus $\operatorname{var}\left( M_{12} \right)=\frac{1}{2n}$.
