Maybe it's prompting to go to "Internet Accounts" in settings to enter your password? Maybe you try this and you get a blank sub-window with the spinning beachball of death? For me, it was because I had set Chrome to be my default browser and the integration between macosx "Settings" and Chrome is broken. 

Open Safari preferences and set it to be your default browser while you add the account, then try again to add your Google account in "Internet Account" settings. It will be happy and let you log into your Google account and add your Gmail, Calendar, etc. to the native mac applications. Then, feel free to switch your default browser back to Chrome.