No, it that is not a legitimate warning coming from Norton!
Such warnings come from adware that got onto your computer without your consent. They usually come in a form of an add-on to your browser (like Google Chrome or Mozilla Firefox), or as a malicious script run from a website you visited.
This is called scareware. The purpose of scareware is to deceive a user to a false claim or statement about his or her computer being infected by a virus or some other type of malware. Then, it is suggested to download fake antivirus software that deceptively cleans an infected computer (pure untruth). In other cases, a toll free number (800 number) is listed, being associated with some of the well known antivirus software brands like Norton. An entity behind such 800 number tries to pretend like they’re an official support of that brand, or at least to make some kind of legitimate connection between them and the brand being (mis)used.
The purpose of such illegal actions is to get a user pay a hefty fee for fixing his/her computer for non-existing problem.
Never fall a victim of such scare tactics. Legitimate security companies like Symantec never charge for support that is already included into your security software subscription. They do offer paid support that goes beyond the one that comes with your service, but they never lure you into it.
More about scareware is described in our Fake Antivirus Scareware article.