Probably not, but exactly why is somewhat complicated. For the sake of example, suppose we drop a proton into the black hole. To an outside observer, once the proton has passed the horizon, it is part of the black hole. There isn't an experiment that we could do from outside the black hole to measure the proton individually anymore. Even within the horizon, it would be difficult to make measurements because light paths are always infalling toward the black hole. Say we use some sort of extremely high frequency laser to make measurements. If we are further from the black hole than the proton, then we can direct the laser at the proton, but the light that bounces off of the proton will never reach us. If we are nearer to the black hole, then our laser will not be able to reach the proton.
If we could somehow solve the measurement problem, then we would find that as the gravitational field strength increases as we approach the black hole, the strong force between the quarks in the proton is also increasing. So the proton is becoming more tightly bound. By the time the gravitational and strong interactions become of the same magnitude, our description of gravity in terms of general relativity is completely invalid, so we can't definitely predict what would happen. Long before we reach that point however, a human would have died, as well as any experimental apparatus would have been crushed by the intense forces.
Finally, the notion of a free quark implies that we can somehow separate the quark from other quarks. When the proton falls into the black hole, the quarks are individually being forced toward the same point, so we can't separate them. It's best to think of the black hole structure as a soup of particles with intense interactions between them. There's probably no sense in which we could describe any of them as free.
