How Can the Quantization Noise of an ADC be Measured using the Digital Output?
An ADC requires an analogue input signal to convert. Something must be known about the input signal, in order to evaluate the ADC. Depending on the type of application, there are several signals that could be used.

The most popular signal is a pure sinewave, covering most of the ADC range. Quantisation noise will tend to produce a flat broadband noise floor, for an input signal that is uncorrelated with the sample rate. Measuring the power in the signal, and in various definitions of 'not in the signal' allows one to separately estimate the quantisation noise (broadband noise), the distortion (power in the harmonics), and for pipeline ADCs, how well the different stages overlap one another (by tracking the behaviour of spurious signals as the input amplitude changes). If all the non-signal power is lumped together, and this power equated to a model of a perfect ADC with quantisation noise, then we end up with the ENOB figure, or Effective Number of Bits. As you're evaluating noise and distortion separately, it doesn't necessarily need to be a low distortion signal for assessing quantisation noise, but it does need to be low noise.

A signal I use when evaluating an ADC that needs to read a DC signal, that I've not seen described in the literature, is that from a slowly discharging capacitor. Obviously the quality of the input signal is paramount when using it as a benchmark to test an ADC, and that from a single large capacitor is going to be low noise, low impedance, floating so free from power supply noise, and very predictable. I make the rate of discharge low enough so that there are at least a handful of counts at each output level. This allows me to check for noise, differential linearity, monotonicity, missing codes, as well as the basic quantisation noise, though it's not useful for drift or overall linearity. With care and very large capacitors, you can characterise right down into the noise.

A third signal source that can be used is that from a 'calibrator', which is basically a precision DAC in a box, with a certficate of traceability from a test laboratory. Needless to say they are expensive, and you're limited to inferring accuracy only up to their certification.