Assuming the motherboard and BIOS will support it, can I change from a dual core processor to a quad core without Windows 7 giving me any grief?
Yes you can. As you say "Assuming the motherboard and bios are all ready to go" if this condition is OK, I mean, if your processor uses the same socket and a compatible chipset you could not only upgrade from a dual core to quad core but the other way as well, downgrading from quad core to a dual core processor.
Both dual core and quad core systems would very likely use the same SMP kernel, and be unlikely to give you trouble due to that. If you wish to play it safe, you may want to run sysprep /generalize to make it a generic, non specialized system before you do the processor swap.
Other issues such as activation may still need to be taken care of, but this should allow the system to be bootable without driver incompatibilities.
For a cpu change there should be no problem, since there aren't any drivers involved (which Windows 7 would get from the internet anyway). Maybe it will ask you to re-activate... but that depends on what has changed, not only now but in the past since your last activation.