I did an experiment, -- restore a copy of backup from production to test server, then do a log backup for the test server database, then restore again from the database itself on the test server, in the restore wizard it shows both the full backup from production plus the log backup.
So this answers I did not need to do a full backup on the test server database after restore.

Somehow I remember I read somewhere it is a good practice to do a full backup after I restore a database from another server.

Thanks,