Yes.
Normally, when you apply some base current to the transistor, it turns on and collector voltage falls. With the capacitor in parallel with the transistor, the falling collector voltage induces a current I = C dV(b-c)/dt between the collector and base. Which means: 1. some of the collector current is diverted to the source, and 2. the source must be able to deliver this current, otherwise it's not going anywhere. The overall effect is to turn it into a transconductance integrator, i.e., Vout = integral(Iin dt).
Other ways to look at it: for a step input, the output is a ramp; for a ramp input, the output is a parabola; for a sine input, the output is cosine (a phase shift of 90 degrees); etc.
The integral is imperfect, only holding true while the transistor is active; a negative step will turn the transistor off, and the only current flowing is the collector load current, which limits the collector-rising slew rate. (In this amplifier, the current comes from an active current mirror, so will be complementary to the input signal, which helps.) The transistor can be turned on quite hard, but no harder than it is ultimately capable of, limiting falling slew rate. (In this amplifier, the current is limited by the input stage bias current.) There is also a feed-forward term (a zero in the frequency response), due to the capacitor itself -- at frequencies higher than the transistor can respond to, or for voltage changes where the transistor is off, the transistor isn't relevant and there is simply a capacitor between input and output, coupling them together directly. That is, a step change of say -0.5V at the base will give an output step change of about -0.5V. This is most relevant for large (>50mV) and fast changes (fractional us to ns), while integrator behavior follows over a slower time scale (us to s).
The relevance of an integrator to an audio power amplifier, is that it gives a precise output, i.e., low distortion and flat bandwidth, when used as the control amplifier. It has a major downside, that its phase shift is 90° to start with, so the rest of the amplifier can't take up much phase shift before the whole thing oscillates. We can free up some phase margin by connecting a resistor in series with the Miller capacitor. This adds a zero at a controlled frequency, easing the phase shift and allowing greater stability and bandwidth. Compare with this circuit:

C1/C2 and R9/R12 serve as Miller compensation. The bandwidth from pretty ordinary components is in the 10s of MHz (at much lower power too, of course).
Tim