# Appropriate Multiple Testing Correction for Correlation Matrix

I have $$N$$ samples of $$D$$ variables, for which I want to study the correlation. Specifically, I want to know if any of these variables are correlated. I can get estimates and a matrix of p-values in R using Hmisc::rcorr, but from the documentation for this function it seems no multiple testing correction is applied. A Bonferroni correction, dividing my threshold by $$D(D-1)/2$$, seems too conservative, since intuitively it doesn't seem like I really have that many independent tests (is the correlation of A and B, and B and C, and A and C, really 3 independent tests?). What would be a more appropriate strategy?

Let $$(Y_{i1},...,Y_{iD})$$ be a random vector, and $$i =1 , .., n$$. Assume $$(Y_{i1},...,Y_{iD})$$ and $$(Y_{j1},...,Y_{jD})$$ are independent for $$i\ne j$$.
Model 1: $$\left(\begin{matrix} Y_{i1}\\...\\Y_{iD}\end{matrix}\right)\sim N\left[\left(\begin{matrix} \mu_1\\...\\ \mu_D \end{matrix}\right),\left(\begin{matrix} \sigma_1^2&...&\sigma_{1D}\\...&...&...\\ \sigma_{1D}&... &\sigma_D^2 \end{matrix}\right)\right]$$
Model 2: $$\left(\begin{matrix} Y_{i1}\\...\\Y_{iD}\end{matrix}\right)\sim N\left[\left(\begin{matrix} \mu_1\\...\\ \mu_D \end{matrix}\right),\left(\begin{matrix} \sigma_1^2&...&0\\...&...&...\\ 0&... &\sigma_D^2 \end{matrix}\right)\right]$$