hi Anuchit

1. what additional role that need to be assigned to Assessor role if the staff manager assign Assessor to update his staff competency?

Not sure what you mean by Assessor role - there doesn't seem to be a standard role called assessor to modify, is this a role you created

2. When will assesor be notify by Totara? Is there Task notification? and what is the sample notification message that the assessor will be notified?

What is the assessor being assigned to - a learning plan, appraisal, feedback 

3. on Learning plan - competency of the leaner page, the competency status that was updated by assessor is showing or not? 

The permissions only apply to managers and learners in learning plans.

4. Where is the help documentation about competency update through assessor located?

https://help.totaralearning.com/display/TL10/Managing+Learning+Plans

Can you take some screenshots to help explain where you have assigned the assessor as not sure what you are trying to do,