One quark would never be created in isolation (the exception to this is virtual quarks, but they can never be directly observed). The deconfinement of two or more quarks requires a minimum energy needed to create an equal number of quarks. So if one quark in a pair gets pulled into a black hole and the other escapes, the black hole loses exactly as much energy as is needed to give the escaping quark a new companion. So quark deconfinement is by definition quark creation. As your very cool gif points out, puling quarks apart is like pulling apart a rubber band, except that the energy needed to break the rubber band leaves you with two new rubber bands in its place instead of two rubber band halves.
But in a scenario with enough energy, it’s in theory possible to create what’s known as a quark-gluon plasma. Again, you will likely never directly observe quark state matter directly. The quarks are what is sometimes referred to as asymptotically free. As it immediately cools, it undergoes a phase change and all deconfined quarks join into pairs or trios or, theoretically, even more complicated quark structures. This is a gross oversimplification, but you can think of it as a momentary condition where the energy density is so high that the quarks have trouble forming bonds, because they’re so “hot” that their motion immediately pulls them apart.
This of course can’t last long, but it’s thought that it plays a role in current best models of the early universe. It may also be possible to duplicate it briefly in particle accelerators. Even though the quark matter itself can’t be directly observed, it can be indirectly observed by it’s products. If you think of a normal nuclear reaction as a nice clean orderly rearrangement of quarks, QGP is more like an internal combustion engine, where heat jumbles things up and all sorts of exotic crap comes out.
Hope that helps. Definitely stealing that GIF 
