Not easily. Unlike the native Google Forms where you can force a user to be logged into in order to submit a form, using the Google Forms plugin cannot enforce the login requirement.
When you use the native Google Form there is a direct connection between the client browser (person entering the data) and Google Drive (where the data is submitted). When using the plugin, the web server hosting WordPress essentially becomes an intermediate point between the end user filling out the form and Google Drive where the form is eventually submitted.
Essentially what happens is the form data is submitted to the WordPress server where some processing is done on it then the data is re-submitted to Google Drive. One of the limitations is the inability to limit responses to one per user and the same limitation is why a user cannot update their response like is possible on Google Drive.
However, you can require the user to be logged into WordPress to submit a form and you can include their username in the form submission. So you could detect duplicate submissions in your results. The limitation here being your users need to be logged into your WordPress site.
