Obviously, regular computers can't generate random numbers on their own, since they're inherently systematic machines. Would quantum computing be able to run a true RNG without a seed based off user input (or any other external variables)?
Actually, today's computers can generate truly random data on their own, and many in fact do. The random data is produced as a byproduct of the physics of the components, not as the product of a given algorithm, so it necessarily has to be implemented in hardware. But the hardware is readily available.
The actual mechanism used to harvest sources of entropy varies by component, but a simple-to-understand mechanism is the analysis of noise generated at the avalanche breakdown of a P-N junction of a diode or transistor. The noise from such a cascade is actually the amplification of the random movement of electrons at the junction - essentially amplifying the randomness of quantum mechanics to the level where it can be read by traditional electronics.
Obviously this isn't the only mechanism you can use to harvest randomness in today's computers, but it's an easy one to understand. In fact it's worth pointing out that the Intel's recent Ivy Bridge architecture introduced a new instruction: RDRAND, which yields the output of an on-chip TRNG. IEEE Spectrum has a detailed write-up of how it works.
Yes, quantum computation allows the generation of truly random numbers, and the operations necessary are so simple companies like id Quantique are already selling quantum random number generators. It is even possible to generate random numbers in a way that proves to the person generating them that they are random (via a violation of Bell's inequality) but this does need a short seed for the proof to be complete (though the numbers are random anyway). Unfortunately commercial systems are not that sophisticated just yet, and so produce random numbers in a way that is difficult to test.
Besides what tylerl says, there's another reason why quantum computers aren't necessary for this. The main difficulty today with implementing quantum computers is to make it be able to do computations on many qubits together. We already have working quantum computers with only one or a few qubits. Those simple systems are enough to produce true randomness of the kind you have requested. In fact, the technology in the quantum cryptography hardware that are already commercially available is enough to produce random bits.