The quick answer to your question is no... PCI cannot be converted to PCIe. The easiest (in fact, really the only way if you wanna use that graphics card) is to buy a new motherboard. I can't recommend or link anything for a new motherboard as I do not know what the VA-10 abit motherboard is. If you know your processor, then I could help you out a bit more.
Also, I'm just curious, what's the video card that your friend gave you? As in, are there any numbers on it? like gtx 560, or gtx 270 or something. Cuz the only info you gave is the company that modeled the whole card and the fact that it's GeForce which means that it should be a decent card.