Let's say I have some ideal generator or battery that will keep two different constant charges C1 and C2 on each terminal.
That's not how voltage sources work. The amount of charge isn't constant. Think of a water hose. You can get water of a certain pressure from the hose, but not because there's a fixed amount of water in it.
Now the battery may have some amount of capacitance. And you can calculate the exact amount of charge that will be on the terminal due to the voltage and that capacitance. But the specific amount is almost completely irrelevant to anything else.
When I measure the voltage using a voltimeter, I seem to be measuring the difference in charge between those points
The voltmeter isn't counting the charges. It can't tell how much is in one versus the other. It is comparing the potential (or pressure) from the charges.
To go back to the hose analogy, imagine you have a tank of water attached to the hose. Based on the pressure of the water coming out of the hose, you can guess at how tall the water in the tank is. But you can't guess the quantity.
Further, you can't tell if the pressure is just from water flowing out of a tank (which would be similar to a capacitor), or if water is being forced by a pump (similar to a voltage source). Your voltmeter is measuring the difference in electric pressure from the two terminals.
Is it possible then to compute volts as a difference of charge over time?
No. Volts don't normally have a time component anyway. For particular devices (like something with a known capacitance), you can map a particular voltage to a particular amount of charge. But that relationship doesn't hold for a battery or a generator. They don't have a fixed amount of charge.
I mean, an excess of electrons in terminal A of a generator and a lack of electrons on terminal B is what makes electrons want to move from A along a circuit and reach B, right?
No. The terminals are just part of the circuit. They are not any more special than the other wires that join up to the generator. The generator voltage doesn't cause charge to accumulate on them to any appreciable amount.
To return to the water analogy once again, imagine we have a water pump hooked up to a fishtank. We see some water is entering the pump on one side, and some water is leaving the pump on the other side. It would be possible to see how much water is in the inlet pipe and in the outlet pipe, but that wouldn't tell us why the water is moving. The water is moving because the pump is creating a pressure difference from one side to the other.
Further, imagine that we disconnect the tiny 5W pump and replace it with a 200W pump. This one is much more powerful and creates a lot of extra pressure. But the amount of water in the inlet and outlet pipes is the same as before. We don't measure the pressure by examining how much water is on both sides of the pump, or in some component of the pump.
The electric generator is doing the same thing, only it's with electric charge instead of water. If you were to count the charges on the terminals with the generator developing 5V and with it developing 100V, the count would be almost identical.