um actually AV's actually can bring in viruses also. Thats how ppl that make these AV make money. U install the Av it will then tell u eventually u have viruse and to upgrade to a newer version. This is partly cos the ppl making the Av's put viruses into their own products so u have to spend more money to upgrade. Sorry if this make u more paranoid or u dont believe me.
Such a business strategy would fall foul of numerous international laws in relation to computer security, privacy and extortion; and would result in numerous lawsuits and sanctions that would effectively destroy the company.
However, there are many viruses that install fake anti-malware products, often disabling any real ones you may have and forcing them to run whenever you try to manage your computer's security instead, which then do exactly as you described. They do not actually scan the PC at all, but instead fake a scan that then tells you you're infected, and demands money to deal with the issue - often installing other malware in the process. These "scanners" are usually not consciously installed by the user, and are not legitimate products. If you find your PC suddenly running an unfamiliar product which claims you have viruses - ignore anything it tells you, and look it up online (perhaps on a different device) - there are usually guides explaining what they really are, and how to remove them.