I am getting a monitor which supports full HD, and my pc specs aren't good enough to run games at high setting at 1080p resolution............. Does the quality of the picture really decreases a lot if I dont run it on native resolution ??
More aboutlcds native resolution
It decreases yes, how much is an individual thing as some people don't notice it as much. Try it on your current monitor and see for yourself what it looks like.
Setting anything at less than native resolution will have an impact on image quality. It will be slightly different between different models. Some LCD monitor can handle it better than others, but it's not like it will make games unplayable.