Safari doesn't suggest new password

Safari does not, "suggest" a password unless you have Keychain selected in System Preferences > iCloud.

If you are not using iCloud Keychain, when you create a new password, Safari will ask if you want to save that data. When you do this, a new keychain is created in Keychain Access located in HD > Applications > Utilities.

In the future, you can save yourself time and effort by writing down your passwords then storing in a safe place.

Keep in mind, not all website support the password autofill feature.

Try this ...open Keychain Access then select Passwords on the left.

Type Gmail or Google in the search field top right corner of that window. If no keychains appear, then you need to change your password manually.

First, go to Safari > Preferences then Autofill tab. Make sure: User names and passwords is selected.

Then create a new password > Change your Google Account password - Gmail Help

When you create a new password, cliick Yes when prompted to save that data to a new keychain. And write that password down.