Fully managed PostgreSQL and MySQL is now GA. This means you can use them in production.
Check the Azure datacenter region availability: https://azure.microsoft.com/en-us/global-infrastructure/services/
User Feedback or feature request: https://feedback.azure.com/forums/597976-azure-database-for-postgresql
Last but not the least is your beloved documentation:
Migrate your on-premises data to Azure using free migration tool from Microsoft https://azure.microsoft.com/en-us/services/database-migration/