Suspicious Apple Security Alert

No it is not legitimate. It appears from your screenshot that a website is showing that to you. Usually those are found on less than legitimate websites where they are paid by these companies so you will download their software or worse to get your account information after calling them to compromise your account. If you are unable to navigate to a different page, then quit Safari and open it again while holding the SHIFT key.

great advice so far. also look at this:

Stop unwanted Notifications - Apple Community

and this:

Phony "tech support" / "ransomware" popup… - Apple Community

Scam!

The message is definitely not from Apple. The supposed Apple Support phone number is certain to be the phone number of criminals who would rip off your wife while pretending to be "Apple Support."

U.S. Federal Trade Commission – How To Spot, Avoid, and Report Tech Support Scams

"Tech support scammers may try to trick you with a pop-up window that appears on your computer screen. It might look like an error message from your operating system or antivirus software, and it might use logos from trusted companies or websites. The message in the window warns you about a security issue on your computer and tells you to call a phone number to get help.

…

If you get this kind of pop-up window on your computer, don’t call the number. Real security warnings and messages will never ask you to call a phone number."

Recognize and avoid social engineering schemes including phishing messages, phony support calls, and other scams - Apple Support

beagletale wrote:

My wife keeps getting this security alert on her Mac. I don't trust it. Is it legitimate? Is there anyway to permanently get rid of it.

That one appears to be a pop-up message – as opposed to a Web notification.

There are controls in browsers to help block pop-up windows, but the only surefire way to get rid of this type of pop-up is to keep in mind which Web site abused your wife's trust in this fashion, and avoid visiting it again.

Scam: Phony "tech support" / "ransomware" popups and web pages - Apple Community

If this pop-up is on a web browser then ignore it.