This link will show the current PostgreSQL versions and what is available: PostgreSQL versions shipped with Omnibus GitLab | GitLab
Technically for your version you could move it to PostgreSQL 13.3 as this is included. The default however is still 12.7 until Gitlab decide to make PostgreSQL 13.3 the default.
This link explains how to do the embedded PostgreSQL upgrade: Database settings | GitLab if you decided to switch from 12.7 to 13.3. Usually though this isn’t needed, and would happen automatically anyway, let’s say if Gitlab decided to make 13.3 the default for when they release Gitlab 15.0 for example.
Also see this forum post: PostgreSQL 13 message but pg-upgrade doesn't upgrade - #5 by iwalker
As to your when question, there is no info on that right now as Gitlab haven’t shared when 13.x would be the default.