How can I interconnect two doorbells operating on different voltages?
I agree with the other answer that it's very possible they'll both work on 6V or both work on 9V.  If that is the case you could wire it so both doorbells are in parallel off one battery.  There may be an issue if the battery doesn't have the power to run both doorbells.  

You could use a relay as suggested in the comments.  With the relay coil in parallel with the 9V doorbell, and the contacts switching the 6V doorbell, you'd be done.  You'd just need a relay with the proper rating, and again the relay coil will take away a little of the power available to the 9V doorbell.  

You could use surface mount DPST momentary switches for the doorbells, if you can find them, but you'd have to run four wires to the doorbell.  

You could get a relay with two pole contacts and put another battery / power source in series with the doorbell switch in series with the relay coil, and have one set of contacts switch the back doorbell, and the other set switch the front doorbell.  

You could install a PLC or home automation system that uses the doorbell as a digital input and the two doorbells as relay outputs, you could do all kinds of crazy things.  You could get a dog that barks a lot or you could put up a sign that instructs people to knock really loudly.  

Of course all these things are harder than just buying a system made for one switch and two chimes, which are readily available, but what fun would that be?