The motherboard has a PCIe x16 slot, so the 670 will fit physically. However, this is a Gen 1 PCIe slot, which is quite slow compared to the modern Gen 3.0 slots. It probably won't be a huge issue, since no modern card shows signs of bottlenecks on the PCIe slots, but it's worth considering.
However, the DQ965GF is a pretty old motherboard, and the CPU on it is probably pretty old as well. I your place, I would rather save up some more and try to upgrade CPU, motherboard, ad graphics. Otherwise, you will not be able to use the 670 to its full potential, since the CPU will impose a heavy limit on what you can play on the machine.