Sorry, but I'm not sure I understand exactly what you're trying to do.  When a quick incident is created, the owner is set by default to the worker who is creating the ticket. 

Are you wanting to autmatically set the contact to the person who is creating the ticket?

Or are you wanting to have the ownership of the ticket change to whichever contact you select?

In the settings of the quick incident you will have to do a set.
The field names you could use are picked from the view workitem_current_view of the Helpdesk database.

Contact_id
Owned_by_worker_nt_id

but I don't have an Helpdesk server in front of me as by the field name givent to the owner it seems it should be a contact but also a worker which will limit your possibilities and might not fit your needs.

Thanks