How do I mark the payment as completed without sending an email to the client?

how to I "complete payment" without sending them an email?

You have to delete your autoresponder email in the form to do so:

https://www.jotform.com/help/132-How-to-Delete-Notifications-Autoresponder

You can re-create it later if you wish to send an email again: https://www.jotform.com/help/26-Creating-a-Form-Autoresponder.