I have an EPM7032QC44-12 device. I need to copy what's on it in order to re-program new dev'. What S/w tool do I need, will Quartus ii Max+PLUS ii support this and does this device support JTAG for 'off' board programming within a programming adapter?

If indeed the part is the EPM7032QC44-12 then it is NOT jtag programmable, and you will need either a third party device programmer that can handle this part, or one of the old Altera Logic Programmer units with a device adapter card. Described here: https://www.intel.com/content/dam/www/programmable/us/en/pdfs/literature/ds/dspghd.pdf

You MIGHT be able to find a third party hardware house that has the capability to program these devices (7000 series, non -A/-AE/-B/-S parts) but this is really old school stuff now.

I personally have a EEtools (www.eetools.com) TopMaxII device programmer which has the capability of programming these series of parts (in this case with a custom TQFP44 to PLCC44 adapter) so if it is important enough to you to do this you might see about buying one of these. Software runs on a modern Windows 7/10 PC, does not require Intel/Altera software to read/program devices).

Good luck to you you are working on an old piece of hardware.