Visiting a website I'd expect any interaction to be with that website, so I assume this is some site where you want to use a Google login - if that's not the case then what does "Google asks me" mean?
I assume that this is a problem with a particular site, so can you sign into the same website from a different device, or using a different browser? If using a different browser works then the problem is between that site and the browser you are using (unless it occurs with any website that allows you to log in via Google, in which case it's between the browser and Google's login system). If it won't work either using a different browser or a different device then the website is simply broken. Do you mind saying what website it is, in case someone wants to investigate the problem?