The title says it all…
My new emachines computer is really slow and I’d like to upgrade the video card and I don’t really feel like going out and paying money, and I have a 1 year old (I’m pretty sure it was on sale for 600$) HP computer (very fast) that is broken and I’d like to know if I can switch out that part to improve my new computer

Yes if it fits in the motherboard and you can download the drivers.