You should be able to set Firefox to automatically log you out of your Google account, although it won't apply to just Google. You will be logged out of every account if you do this.

Go to the Firefox settings or type about:preferences in the Firefox location bar. Go to the Privacy & Security tab in the sidebar and find the History section. These are the settings that are required to make this work:

• Firefox will use custom settings for history
• Check Clear history when Firefox closes
• In the associated Settings... button, Active Logins needs to be checked

Hope this helps.

You can use "Clear history when Firefox closes" to clear the cookies. If you leave tabs open when you close Firefox then cookies are stored in sessionstore.jsonlz4 as part of the session data. You can modify this pref on the about:config page to prevent Firefox from saving these cookies.

• browser.sessionstore.privacy_level = 2

You can open the about:config page via the location/address bar. You can accept the warning and click "I accept the risk!" to continue.

• http://kb.mozillazine.org/about:config

A detail like a website remembering you (log you in automatically) is stored in a cookie.

You can use these steps to make a website recognize and remember you:

• you can create an "Allow" exception to keep specific cookies, especially in case of secure websites and when cookies expire when Firefox is closed.
• Options/Preferences -> Privacy & Security
  Cookies and Site Data: Exceptions