According to a BMC Bioinformatics article, WGCNA can be used for finding clusters (modules) of highly correlated genes, for summarizing such clusters using the module eigengene or an intramodular hub gene, for relating modules to one another and to external sample traits (using eigengene network methodology), and for calculating module membership measures. However, I could not find any information on how to find the correlation between genes that make up a module.
Regarding your question about how to find a relationship between genes that are negatively correlated, one possible solution is to use a combination of signed and unsigned WGCNA (csuWGCNA), which combines the signed and unsigned methods and improves the detection of negative correlations. According to a bioRxiv article, csuWGCNA was applied in 14 simulated datasets, six ground truth datasets, and two large human brain datasets.
Finally, you want to draw networks having edges, red representing negative correlation, and blue showing positive correlation. One possible solution is to use Cytoscape, which is a free and open-source software platform for visualizing molecular interaction networks and biological pathways. According to a GitHub repository, you can export the network files for visualization in Cytoscape and correlate the modules with external traits (discrete type), correlate the modules with external traits (continuous type), and further investigate particular module-trait relationships. You can also visualize the networks using scatterplots or bubble plots.
Reference: https://bmcbioinformatics.biomedcentral.com/articles/smash karts10.1186/1471-2105-9-559 and https://github.com/Lindseynicer/WGCNA_tutorial
To find the positive and negative correlations between genes in a WGCNA module, calculate the correlation coefficients between gene expression profiles. Visualize the module network using a tool like Cytoscape, using blue edges for positive correlations and red edges for negative correlations. You can visit: https://github.com/Lindseynicer/omegle/WGCNA_tutorial