This article explains how to create a new Apple Account --> How to create a new Apple Account - Apple Support
Once on the page, select the kind of hardware you are using.
If you are doing as instructed in the article, it is not working, and you have checked your mail application, then contact Apple. Refer to this page for Apple Support features ➔ https://getsupport.apple.com/products
Select from the presented options until you find a solution for your issue, or see if there is a chat or phone call contact method offered lower on the page under “Get more help” (you sometimes have to narrow down the options multiple times before this is shown). If you do not see your issue, keep experimenting with any series of selections until you reach one that offers a chat session or a telephone call and get the representative to redirect you.
Have you also tried logging into Gmail with a web browser? That would at least cut out possible issues with using a Mail application of some kind.