Hi,
To log in to my.avast.com you need to enter an email address (as user name).
If the current email address (used for your Avast account) is the as the google.com email address then
you can simply login using the google option.
If they are different then you need to add the "new" google email as an additional email (and Set as primary).
Please see this guide:https://support.avast.com/en-ww/article/Account-FAQ#idt_090