One common and relatively simple method of looking for evidence of a whole genome duplication is to estimate the rate of synonymous substitutions (Ks) between duplicated gene copies and then plot the distribution of Ks and look for peaks in the distribution. The logic of this is that if many genes are duplicated at once (which obviously occurs during a WGD), there should be a relatively large number of paralogous gene copies with similar Ks values, but if genes are simply being duplicated at some low underlying rate, you would not expect peaks in the distribution.

There are many papers that discuss this type of method (and its limitations). Here is recent one:

https://academic.oup.com/gbe/article/10/11/2882/5100828