I am familiar with the persistent and non-persistent XSS. I also know about Same origin policy that prevents/restricts requests originating from one websites page to go to anoth
Typically no.
A non-persistant or reflected XSS attack exploits input that is echoed back as page content without proper sanitization, without persisting it. The injected script will seem to come from the exploited domain in both cases.
For example if you do this in PHP: echo $_GET['param']
and send a link to the page to somebody containing
?param=
it is a non-persistant XSS attack, and same-origin policy has nothing to do with it.
Same-origin means that you cannot directly inject scripts or modify the DOM on other domains: that's why you need to find an XSS vulnerability to begin with.