If the original image is at 100% zoom (so each pixel on the screen represents exactly one pixel from the original image) then there will be no loss of quality.
If the image is at higher than 100% zoom, then there will be aliasing present in the resultant screenshot which is technically in the original image, but is not visible to the same degree at the correct zoom level. 'Fixing' this aliasing with an anti-aliasing filter will technically not accurately represent the region you are extracting (though it may do a better job than the original image if it's mostly linework instead of things like gradients).
If the image is at less than 100% zoom, then you lose quality because of the interpolation algorithm used by whatever image viewing software you are using.
All three cases assume you are using a lossless encoding format (such as PNG, or GIF, or even an uncompressed image format) for the resultant image. If you use any form of lossy image compression (for example JPEG, or the new HEIC codec), then quality will be lost regardless of the scale of the original image. Also note that this assumes that you're taking the screenshot digitally, and not with a camera pointed at the screen (that will always reduce image quality, even if you've got a nice camera that saves lossless images).
It is probably worth noting that this is a really inefficient means of extracting a particular region of an existing image. You will almost always have to edit the resultant screenshot to pull out the part you want, but doing so requires the exact same techniques that you would use to just pull the region directly out of the original image. The process itself is really trivial with any decent image editor such as Photoshop or GIMP, just use the select tool to outline the region you want to extract, crop the image to the selection while resizing layers, and export the result under a new file name.
3Zoom to 100% before taking screenshot. – Sandeep – 2018-06-21T15:39:30.527