Can I put DDR3 4GB RAM on a Core 2 Quad Q6600?

It depends on the motherboard, in this case. Some LGA 775 motherboards use DDR2 RAM while others use DDR3 RAM. You cannot use DDR3 RAM on a motherboard that uses DDR2 RAM.

And the Intel 3- and 4-series chipsets designed for the Core 2 Quad Q6600 support either DDR2 or DDR3 RAM, but not both at the same time. It depends on your motherboard. And these are the last Intel platforms not to have their memory controller integrated into the CPU package, but instead located externally on the motherboard.

At any rate, that platform is now completely obsolete. It is no longer supported anymore with even security fixes, let alone newer OSes.