Is there a quantum NC algorithm for computing GCD?
First of all, there is a formal definition of "quantum-NC", see QNC on the zoo.

GCD is indeed a good candidate for a problem that could be shown to be in QNC, but it's not known to be in NC. However, finding a QNC algorithm for GCD is still an open problem. 

The feeling for which this is believed to be true comes from the fact that the Quantum Fourier Transform can be done in QNC.

Reference: Conclusion section of "R. Cleve and J. Watrous, Fast parallel circuits for the quantum Fourier transform", arXiv:quant-ph/0006004