Find which account SQL Server service is running BACKUP under

BACKUP will access the UNC under an impersonation context. This is a 'double-hop' situation and requires Kerberos constrained delegation to be configured.

This is already answered at https://stackoverflow.com/questions/18749224/restore-database-from-a-shared-folder

If you run a BACKUP query under a certain Windows/SQL Server account, it is actually the SQL Server Agent account which "does" the BACKUP and needs full control permissions to the UNC path. So you must be able to allow that account (or Everyone) have full control permissions to that path. To find out which account SQL Server Agent runs as, query sys.dm_server_services. It requires that you (or someone else) have VIEW SERVER STATE permission on the server.

Do you have the option of creating a folder on your local computer and create a share on that and grant full authority to everyone and then back up to that share? - you didn't indicate how big the backup might be