Will this graphics card work for my computer?

The GTX 960 graphics card that you linked has a system requirement of 400 watts. The HP system that you linked only has a 300 watt power supply. Also you would need to measure the space available inside the case to determine if it has enough space. Pre-built systems like this very often do not use standard size cases.

So in answer to your question, you would need to upgrade the power supply and determine the space inside the case for the card.
 
The card you have selected will fit. However, as has been mentioned above, the PSU you have is not really powerful enough to run it safely. I would consider running a GTX950 though. However again, you need to check that your CPU has a PCI-e 6 pin power feed for your GPU. The GT 520 you have is a very low power use card, around 30W, and it does not need separate power. If you lack the power feed, the GTX 750ti (some models only) like the EVGA GTX 750ti Suplerclocked 3753 I use in this system, is about the most powerful GPU that you can install without a PSU upgrade.
 
Depending upon your budget there are several options. For example if you upgraded the power supply to 650 watts. You could upgrade the graphics card all the way to a GTX 980 Ti if you so chose to do so. In other words the power supply is the limiting factor.

.A GTX 750 Ti would probably work for you . But just realize that it is an entry level card. Is that enough graphics power for your purposes?
 

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