The US firm says it will bring up a banner highlighting the danger when users are logged into their Google Account.
An attached link will suggest actions to protect their data from being hijacked such as the use of tougher passwords and suggested system updates.
The search company would not reveal exactly what would trigger the alerts.
It only said that it would be based on the belief that a user might have become the target of malware or phishing – requests for some of their personal details from a party pretending to be a trustworthy source.
"You might ask how we know this activity is state-sponsored," wrote Eric Grosse, Google’s vice-president of security engineering on its blog.
"We can’t go into the details without giving away information that would be helpful to these bad actors, but our detailed analysis – as well as victim reports – strongly suggests the involvement of states or groups that are state-sponsored."
A recent report by Reporters Without Borders flagged up several instances of what it said were state-backed attacks over the previous year.
It said one case involved users in Belarus who had tried to log into a local social network but were redirected by their internet service provider to sites containing malware.
It added that attempts to "phish" for social network usernames and passwords had been reported in Syria and Iran.