I'm sorry, at first I did not understand your question.
You want to change the default name used when saving a document based on your template.

By default, Word will use the contents of the Title document property for this. If there is nothing there, it uses the words that start your document.
A simple thing to do is to insert the Title Document Property Content Control in your template.
Insert > Quick Parts > Document Property > Title

You can set this in your template and that will then be the default name, which can be changed when you use the template by changing the Title in the document.