Can a Computer Motherboard Speaker/Buzzer output voltage switch a relay?
A motherboard buzzer takes a lot less current to operate than a relay winding. So it could work for a bit, but it might burn out the driver. In other words, it's the current drive that matters.

You're much better off using the speaker output to drive a transistor (BJT or MOSFET) that switches the relay. Transistors require very little input current to operate. They can also level shift, i.e. drive a large output voltage swing with a smaller input voltage.

Remember the reverse-biased diode across the relay winding, which clamps the voltage when you turn off the relay. Otherwise, the relay winding voltage will head for -4 V or -12 V (the negative of whatever voltage was on it when it was on) and not stop until some part of your chip looks like a forward-biased diode. It's an inductor, which can change voltage instantly but not current.