Responsible cannot be changed thought simple tickets view.
You must use Ticket view.

Responsible lists all resources affected to the project.

Sure, I am now using (main) Tickets screen and got the Responsible field there.

I think the values for "Responsible" shows all the Resources. I want to show only selected profile of resources in this dropdown - we call it as "Function Approver". I have an user profile by the same name and a few users are tagged to that user profile. Please help me how do I change the values of Resources to something I want?

Alternatively, can I add a new field "Function Approver" and show list of the specific users?

I think the values for "Responsible" shows all the Resources.

It sholw lost of all resources affected to the project.

I want to show only selected profile of resources in this dropdown - we call it as "Function Approver". I have an user profile by the same name and a few users are tagged to that user profile. Please help me how do I change the values of Resources to something I want?

You cannot on standard version.
This would need specific development.

Alternatively, can I add a new field "Function Approver" and show list of the specific users?

You could add new field with the "screen customization" plugin, but you won't be able to define the expected list of users as this would require creation of a new object. This wouldalso need specific development.

We can do developments for you if you please. Just request for a quote in the contact form.

Thanks Babynus for the response. I will certainly contact you as needed.