You may not be able to use the postgresql DBAAS as the musicbrainz database contains the musicbrainz_unaccent and musicbrainz_collate extensions to postgres.
If you use the google managed database you do not have the ability to add your own custom extensions but can only use the extensions that google provide and would need to request that they add our extensions to the list.
https://cloud.google.com/sql/docs/postgres/extensions
This is not stopping you from creating a vm and running a database yourself but that does not have the advantages of getting google to do this maintenance.