Currently the backup,copy and restore works fine in my server just
backup status not show at report transaction log shipping.
If backup, copy and restore is working fine then its objectivity is obtained and one site to another site is in sync however in case you are unable to see only the backup status, you may check this at individual job level by clicking on job history at primary server and see whether it is successful or has completed with error.
Alternatively, you may use script to check the current status of log-shipping at primary and secondary server:
select * from msdb..log_shipping_monitor_primary
select * from msdb..log_shipping_monitor_secondary
Primary will give you information about backup whereas secondary will have details on both the jobs i.e. copy as well as restoration.
You may try below stored procedure also which will return result however un-formatted:
The scripts work fine but the primary database not show in secondary
server log shipping report if I select
[msdb].[dbo].log_shipping_monitor_primary in secondary server my
primary database not insert at that table.
This is supposed to return rows at primary server and not at secondary server, it seems you are mistake about this table.
sp_help_log_shipping_monitor can be run on both the nodes i.e. primary and secondary and will return result for backup on primary whereas on secondary this would give details about copy and restore.
Hope above helps.