I assume G: is a network share located on another drive? If so, 'G:'
might be pointing to a different path under whatever account the SQL Server
service is running under. Can you run the following command in SQL Server
Management Studio (or SQLCMD.exe):
exec xp_cmdshell 'DIR "G:BackupFull Backup"';
This should show the content of the folder actually used by SQL Server
when performing the backup.