There is no "Flash Player HD" from Adobe. Flash Player will play HD video if the video embedded is HD.
THOSE are the "legitimate" installers.
In Windows, go to: Control Panel>System and Security>Flash Player (32bit)>Advanced to check your version.
In Mac OS X, go to: System Preferences>Flash Player>Advanced.
The current version is 11.9.900.152
If yours is lower than that, use the appropriate link(s) above to update.
If you STILL see the update notices, you have malware producing them, but such malware wouldn't work on a Mac.