HP Pavilion P6-2065UK graphics card?

well, the best way of gathering that information is if you measure the available space with a ruler or something, and compare with the GPU you are looking for Dimensions, but i don't think that is the problem, see, those pre-build computers always come with specs for optimum performance for the current hardware, and because of that they use normally use a Power Supply Unit that will fit the need only, so the problem is 300W psu (the default one in that model) isn't safe for any good mid-spec card, normally they require at least a 400w psu from a good brand, so the point is, do you have the money to exchange PSU and GPU? if you are aware of that, than you can measure spaces on the case, it is a mini tower so probably you will need a very good cable management to fit a big gpu there, best option is going for the gpus with lowest profiles since they are smaller to fit on smaller cases, also make sure a good PSU will fit there.

if you give us a budget we can help you finding good options of PSU/GPU match for that machine. also what is your monitor resolution, and for what you want the gpu? gaming? if yes what games?

Case Size specifications:
Mini-tower

Height: 36.8 cm (14.4 inches)
Width: 16.5 cm (6.4 inches)
Depth: 38.9 cm (15.3 inches)
Weight: 8 Kg (17.6 lbs) - without packaging

well but fit isn't really a problem, if something don't fit there, you can always change your case, or even assembly components out of it.