The documentation for Posgres on Kubernetes seems to be missing the direct instructions on how to upgrade the Operator from version 2.3.0 to version 4.2.0
The R&D team confirmed that, due to a significant architectural change from version 3.0 to version 4.0, that the Operator must first upgrade to version 3.0 first.
Once the Operator has been upgraded to 3.0 then you must download and run a pre-upgrade script.
The script adds a user and the necessary privileges required for version 4.0 to run.