Hello t_m,
Please note that there is no way to change the onboard graphics card on a processor. You can upgrade the system by changing the processor with a better/ newer onboard graphics controller.
Nowadays we have the 4th generation processors that come with the Intel® HD graphics 4200, 4400, 4600, 5000 & 5200. I am afraid to say that since you have a 2nd generation processor your motherboard does not support the new Intel® processors meaning that will be necessary to change the motherboard too.