How to get Pages to automatically save documents as .docx files
Pages does not support saving as a Word .doc or .docx file.  Your only option is to Export your document to that format, which obviously isn't nearly as easy as simply hitting the Save button.

It's important to note that Pages support of .doc/.docx isn't perfect.  If the document has a lot of formatting or review comments, there's a pretty good chance of lost formatting or comments when you export your document.

A free alternative to purchasing Office:Mac is to use Office Online, which will happily run in Safari or Chrome on your Mac.  If this is a short-term need, you can download a free trial of Office:Mac.  If you have an iPad, there's a free version of Word for it too.  I realize that none of these are what you've asked about; I provide them in case an alternative to using Pages will work for you.