Can a coil and a magnet moving together produce a voltage? In other words, if a coil is wound around a fixed magnet and the combination is rotated or otherwise moved, will the coil produce emf? Must the permanent magnet move relative the the coil to produce emf? I recall seeing a demonstration during a physics presentation back in the 70s where this was the case. The magnet and coil were spinning together, producing a few watts. But perhaps my memory is faulty, or I am mistaken that the coil was in fact attached to the magnet.
 A: Have you heard about the Faraday disk generator?

A is a horseshoe magnet,
B a rotating metallic disc
Such device produces a current not only when the disc is rotated but also in the case, the magnet is rotating together with the disc.
What is the mechanism behind the phenomenon of a homopolar generator?

According to Wikipedia  


The motion is azimuthal and the field is axial, so the electromotive force is radial. ... If the magnetic field is provided by a permanent magnet, the generator works regardless of whether the magnet is fixed to the stator or rotates with the disc.


Our conclusion should be that electrons under a centrifugal force and in a constant magnetic field are moving according the Lorentz force, the current is perpendicular to both the movement and the magnetic field. And we have to add that this movement should be accompanied by the centrifugal acceleration of the electrons. Moving the disc and the magnet along a line does not induce any current.
Following the above said I try to design a rotating coil with an attached magnet in such a way, it will induce a current. I see no way, but maybe you’ll remember it, applicating the above said. 
A: Let's qualify the question by ruling out sliding contacts. This excludes
unipolar machine which is harder to explain and IMO the questioner
didn't think about. Then the answer is no, at least if we stay to a
nonrelativistic approximation (slow motions).
As @pinchun wrote

the coil will not feel any change on magnetic flux

To rigorously prove that statement starting from Maxwell equations
isn't trivial. I mean it isn't trivial to prove that induced emf (you'll pardon me if I don't use "voltage" - I think words like that should be banned from physics) only depends, in every situation, only on variation of
magnetic flux through induced circuit. Not trivial, but true.
Another comment I would make about rotating magnet. It was cited by
@HolgerFiedler in context of unipolar machine, but the same argument
can be put forward in other cases too. If we have - for instance - a
stationary circuit and a cylindrical magnet rotating on itself, so that it exactly reproduces at every instant the same magnetic field, it would be wrong
to think an induction will happen because "magnetic lines of force are
cutting the induced wire". Lines of force are not material objects,
attached to the magnet. They are only graphical representations of the
magnetic field at a given instant. It magnet rotates always remaining in the same place, in every point of space magnetic field stays constant, so no flux variation will happen.
A: I think the answer is no.
As @FGSUZ has mentioned in the comment, since there are no any relative motion between the magnet and the coil, the coil will not feel any change on magnetic flux. Therefore, it does not produce a voltage.
Although you said "the magnet is moving, and the movement of the coil doesn't matter", however, I don't think such movement would cause magnetic flux change to the coil. Or maybe you can give an example?
