Azure SQL Database leverages SQL Server engine technology for data backup and restoration. In Azure SQL Database, all backups are handled by Azure, and I believe there is no specific feature to receive alerts for backup failures.
Backup failures are rare, and even if they occur, I believe Azure backup service automatically retries the backup in subsequent occurrences.
The sys.dm_database_backups DMV Returns information about backups of a database in an Azure SQL Database server.
Show list of all active backups for the current database ordered by backup finish date.
SELECT *
FROM sys.dm_database_backups
ORDER BY backup_finish_date DESC;
Additionally, you can monitor the consumption from portal.
For vCore databases in Azure SQL Database, the storage that each type of backup (full, differential, and log) consumes is reported on the database monitoring pane as a separate metric. The following screenshot shows how to monitor the backup storage consumption for a single database.
Hope this helps!
Let us know if you have any further questions or need additional information.
Thank you!