Making the DE1 version of the Minimig core output four independent audio channels would be a fairly simple hack - a rough-and-ready solution would involve adding four Delta/Sigma audio outputs to the code, and sending them to pins on the general purpose IO headers. A resistors/capacitors filter on each output and you're good to go.
(The sound quality wouldn't match that of the DE1's built-in audio chip, however.)
Without checking the schematics I can't be sure, but I should think the original Minimig's core could be hacked quite easily - again with four delta/sigmas - to send the four channels' sound outputs through one of the game ports.
Adding extra channels would be theoretically possible but more complicated - and of course would require new software.