select count(*) from msdb..backupset where database_name = 'yourdb' and type = 'D'

However older backup history can be purged from this table by cleanup operations so unless you never run this purge and just allow the table to grow ad infinitum the info could be wrong. This table has an identity column backup_set_id you could use as an ever increasing number, but it would not reflect the actual number of backups for that database.