Welcome to Justanswer.com, my name is Dustin.I'm sorry to hear about the issue you are having, I will do my best to help you resolve it.
Had you ever successfully logged in to google / gmail on your mac?
If so, I can connect to your mac from here and go through your stored password ***** and help you find the original password ***** else help you change the password.
I am sending you an offer for the screen sharing session now, If youd like to try that accept the offer. Otherwise, tell me you only want written support and we can try to work through the issue via written support.