I don't think so, How many traits are you planning to test?

If your alpha is 0.05, the family wise error rate for 10 traits
1 - (1 - alpha)^num_traits = 1 - (1 - 0.05)^10 = 0.401

or about a 40% probability that for at least one of your significant traits, its correlation with the Eigengene is due to chance and not due to your experimental manipulation. The simplest (and strictest) correction is Bonferroni's method which for 10 traits would mean using an alpha of 0.005 for each test giving you an alpha of ~0.05 for the whole family of tests.

I suppose you could correct the p-values, but to be honest if you have enough samples to be doing WGCNA and the correlation is strong enough to be worth looking at, it will likely have a very very low p-value anyway.

What you want to avoid is a situation where you are using a nominally "significant" p value to justify a weak correlation that likely has no biological relevance to the trait.

My recommendation is that instead of emphasizing p-values, that you look at the modules that correlate most strongly with the traits in your study and try to develop a deeper understanding of how those genes might interact with one another, or otherwise influence the physiology of your system.

Once you've identified module(s) of interest, you can look at which genes in the module are differentially expressed with respect to the given trait using edgeR/DESeq2/Limma etc (even if your trait is a continuous variable). I think testing for significant genes associated with an interesting module will be more robust than the correlation p-value anyway.