Not at all.
Alpha diversity is telling you about the diversity (e.g., richness) within each individual sample. How many species are present? How much of the phylogenetic tree is covered? When comparing groups of samples (e.g., with a Kruskal-Wallis test), you are asking whether these values differ between groups. Does group A have more unique species/phylotypes than group B? A significant P value will indicate that yes, group A is different from group B.
Beta diversity is telling you about how different samples are from one another. This may be phylogenetic (UniFrac) or non-phylogenetic (e.g., Bray-Curtis). This may be weighted by abundance or relate to the presence/absence of individual organisms. You really need to understand what the diversity metric measures (this is true for both alpha and beta diversity). A permanova test is going to tell you whether those differences partition based on some metadata value — is at least one group significantly different from at least one other group. A significant result is essentially telling you that samples in group A are more similar to each other than they are to samples in group B, and vice versa.
So these tests are telling you very different things, and the specific interpretation depends greatly on the metrics that you are using.
I hope that helps!