You can set up a CmdExec job that connects with SQLCMD to the other instance and runs sp_start_job. You would have create a proxy user that has permissions to start jobs on the other instance, and you would select this user as Run As for the job.
Call SQL Agent Job on different instance

I have a SQL Job on a 2008 R2 instance and would like as it's final step to start another SQL job on a SQL 2019 instance. Is this possible?
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AmeliaGu-MSFT 13,901 Reputation points Microsoft Vendor
2021-09-23T06:03:45.86+00:00 Hi MarkSanchez-5028,
You also can try to configure a linked server to Server 2 in SQL Server 1, and use the linked server with executing msdb.dbo.sp_start_job to call the SQL Server job in Server 2 as the final step of Server 1 job.EXEC [Server 2].[msdb].[dbo].[sp_start_job] @job_name = N'jobname';
Please refer to How to Create a SQL Server Agent Job that Calls Another Job and How to start SQL Agent job from Another server for more details.
Best Regards,
Amelia
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