Dell OptiPlex GX620 can it handle a Core 2 Duo Cpu?

Afraid not. It's still in the single core era...
http://www.dell.com/downloads/global/products/optix/en/...

It just so happens the GX620 USFF that I'm using at this moment is happily running a Pentium D 3.4. That's the limit for all of the GX620s, but the Pentium D is, indeed, dual core.

You are correct. I shouldn't have refered to it as being in the single core era. I forgot about the early Pentium D being a dual core. But in my defense, the OP asked about a Core 2 Duo processor. I don't believe the machine can handle that even if it has a LGA 775 socket. Knowing which motherboard the computer has would be helpful.

With stock motherboard, No. If you have an upgraded one then yes it proberaly will.
Even if you done you can pick up a motherboard for a bout £35 that willbe fine with a proper Dual core