Blue-Green deployment is generally not applicable to database servers.
For zero downtime deployments you can deploy schema and data changes in a transaction so the database changes atomically from one version to the next from the client's POV. This can cause a brief delay in both the deployment process and in the application workload as one or the other waits on locks.
But the real reason to perform deployments in a downtime window is to coordinate the deployment across all the application components.
So if you want to deploy changes to a database without downtime, you must ensure that the running version of the application supports both the old and new version of the database. This sometimes requires non-trivial orchestration of application and database changes.