It sounds like you've been told that it will not work.
Physically, given you have a socket 775 mother board, a socket 775 CPU will fit in your socket.
The QX9775 C2E CPU, which is listed on the Core 2 Extreme Processor page, is spec'd to be a socket 771 CPU. I know those two sockets are very similar, but there a couple pin/contact changes, and the indexing notches are in different places. I've never owned that CPU, so I am not clear on the compatibility. The other C2E processors are socket 775 CPUs.
Usually an incompatibility problem is due to the mother boards BIOS being able to support a CPU, even if they use the same socket. The only way to know is to check the CPU compatibility list of the boards manufacture.
I can't find a D945GTP board on Intel's site.
I think this D945GTP Compatible CPU page is the one you were looking for parsec (after selecting the 'discontiunued' radio button). Given your investigations into that CPU and what it says at the bottom of the page I've linked to, the o/p needs to be aware it may be money down the drain trying that CPU. If you look at the compatible boards listed for the C2E CPUs (looking at it the other way around), they're at LEAST 965 chipset boards (and mostly newer than that) but then having not tried it myself, I can't give a definitive answer - my hunch is that it'll be a no however and if it was me, unless I was given the CPU free or had it to try anyway, I'd just get one of the known compatible CPUs listed on that page.
The main reason I can think of to pursue the upgrade would be to avoid software and RAM changes. I believe the 945 series is newer than the 965 series mentioned by Kiwi, but they generally have more limited features and connectivity. The better option would be to find the newest board that is compatible with the Extreme CPU. Features such as hyper-threading would become available with better boards. Changing the CPU model will require reinstallation of software anyway so a totally new and modern system would be preferable in most cases. That is preferable to having a system that seems to mostly work faster.