It will be made available for albums loaded from MB if they have genres set. Let me guide you through this.
First in Picard enable usage of genres in Options > Metadata > Genre. The settings as shown below should do:
Important is to enable “Use genres from MusicBrainz”. Don’t enable “Only use my genres” unless you have tagged all your releases with proper genre on MB yourself. The fallback on album artist is probably a good idea in most cases. Also note the “Minimum genre usage”, that makes sure only the most popular genres are used. I’ll explain below. For starters the defaults should do, you can look into fine tuning if you are not satisfied with some genres you get. See the documentation for details.
If enabled loading a release in MB on the right pane will make the genres available that you can also see on the MB release page. Let’s take as an example this release group on MusicBrainz:
On the right you see it has three genres set (gothic metal, gothic rock, heavy metal). Also note that the genres have been voted on, “gothic metal” got 4 votes, the other two only 1. You can see that If you load one of those releases into Picard it will also show the genres:
If you tag your files against this loaded release the genre tag will be saved to the files.
Now you probably noticed that in this this case only a single genre “gothic metal” was loaded, but the release group had three. That is because the other two have only 1 vote, which is less than 90% (the “Minimum genre usage”) of the most voted genre (4 votes).