According to Faraday's law of induction, volts = -Number of coils in a solenoid * change in strength of magnet / change in time. This doesn't take into account distance or speed, only time. If amps = volts / ohms, and ohms is 0, it seems like amps should be infinity. If there are infinite amps, then wouldn't the alternator just generate a constant voltage without the magnet needing to move at all?
You seem a bit confused on your standing conditions, so we'll go at this step by step.
Let's start with this statement: The magnet does not move
From this, we can easily deduce that $V=0$
Now, $V=IR$, and we have $R=0$ as well. Thus, $0=I\times 0$, and then $I$ can be any value. (In case of a non-perfect superconductor, we have $R \approx 0$, and we can derive $I=0$, since $0=IR$ and $R\neq 0$)
OK, so you got a constant current. What of it? The current will only be able to course forever in the superconductor ring. You cannot try to draw energy off it, because all ways of drawing energy off current increase the resistance. If you add a load coil/etc, there will obviously be resistance. If you try to harness its magnetic field, the fluctuations in the field will induce a reverse current on the coil, acting as a resistance.
You completely forgot the other alternative--volts can be zero instead. Then you can easily see that the current can be any value.