You would need to share the folder on the linked server and grant either the SQL Server service account (if running the backup through SSMS) or the SQL Agent service account (if running as a job) write permissions to the shared folder. Those service accounts would be on the server running the backup not the linked server service accounts. Then you can run something like below.

BACKUP DATABASE dbname TO DISK='\\linkedserver\backupfolder\dbname.bak'