Which alpha and beta diversity metrics are recommended?
Comparing diversity between sample categories
Interpreting alpha and beta diversity
The multiple test problem and fishing for significance
If you have designed an experiment to test a hypothesis, and this hypothesis suggests that a given metric will change in a given way, then you can use a standard test such as Wilcoxon on the metric that was predicted to change. However, it is not valid to systematically try several metrics to check if one of them gives a statistically significant change between groups. This is fishing for significance, which causes a multiple test problem. If you do this, you should adjust your P-values using the Holm-Bonferroni method or other applicable technique.