Let's say I have a photo that is 1000 x 1000 pixels and I use two different monitors to visualize this photo, one with 1920 x 1080 pixels and the second with 640 x 480 pixels. I assume that all pixels will be shown on the first monitor. But what happens to the photo in the second monitor? I read somewhere that the image is simply down-sampled. If I decide to print this photo (say 300ppi), does the quality of the print depend on which monitor I am currently viewing the photo?
You have specified the expected print resolution as 300 ppi, which is what determines the quality of your printed image.
Consider that you can place the image on a portable storage device and provide it to a printing service with the above specification. Without seeing the image, you and the service will have the same expectations of resolution.
The monitor on which you view these images merely presents the image to the scope required (smaller on smaller monitor) and has no effect on the printed output.