Could a black hole pulling on a neutron star temporarily create a quark star? I believe a quark star is a hypothesized star that is composed of quark matter. If I'm correct then an even large gravitational pull than a neutron star has would be required to break down the individual neutrons forming a star made of quarks. For a neutron star to do this it would require so much more mass that it would make a black hole before coming a quark star.
What if a neutron star was caught in a black hole, could it become a quark star as it was being pulled in? 
 A: You asked one question and made one statement. First the statement, for which the answer is possibly no. Afterwards, your question. 
The statement you made was that before becoming a quark star the star would collapse into a black hole. This has not been definitively calculated or observed. So, no, theoretically a quark star could form if enough matter was present, and there was a quark configuration that is stable and does not further collapse or get compressed to its Schwarzchild radius. The problem is that the equation of state of quark matter is not well established. 
It is, however, estimated that normal quark matter can not be stable interacting with a vacuum (I.e., no quark matter on the surface) , so there's only two possibilities. One that the core of a neutron star become quark matter due to the high pressure with enough matter, and two that the material in the quark star is strange quarks. See the Wikipedia article on it. 
https://en.m.wikipedia.org/wiki/Quark_star
Nothing has been observed and confirmed to be more dense than neutron stars, although a couple have been reported. So there's been no observations of one. So it is estimated that if they exist, there are not a large number, relatively, of them. Still, not totally ruled out. 
The mass at which a black hole must form is still not completely determined, it is somewhere between 1.5 and 3 solar mass, and clearly above. Neutron stars hhave been observed in the range of 2 solar masses. Similarly, black holes below about 3 solar masses have not been observed yet. (My numbers are approximate, I could be off some, but there is a range of masses that have just not been observed). So there is still a possibility, but no observational evidence still of any quark star, or a neutron star with some quark core.
Your question then was whether a quark star, or let us say a neutron star with a quark core, could be sustained as one while being pulled into a black hole? For the quark star itself, possibly, but it already exists anyway, and maybe during its getting sucked in the pressure internally could subside some and some of the matter become neutrons. Similarly for the quark core, strange or not. The answer is you'd have to do some pretty heavy calculations, and assume some quark equation of state, so it would depend on that. The evolution of a neutron star as it goes through its absorption by the black hole needs to be numerically calculated over its last few orbits, and then as it gets very close to the horizon. You'd have to do the same with the mixed neutron/strange-quark option, and the quark star itself, with maybe various different equations of state.
There is a site that describes some of the neutron star black hole mergers by using numerical relativity, and it you wanted to try it on various quark equations of state that might be a place to start. It's at 
http://www.black-holes.org/the-science-compact-objects/compact-objects/neutron-stars-and-pulsars, where there are other numerical relativity calculations on black holes and neutron stars also. 
I'm not sure what new physics you'd get if they were found, but we might for instance find out more about quark equations of state, and states in the early universe expansion. 
A: No and maybe a yes. Quark stars formation  from a neutron star in this is hard to conceive.
Yes it might be possible for black hole to collapse a neutron start into a quark star hybrid exotic star under different circumstances.If a small but significantly massive black hole collided and acted like a strangelet.
A shining example hybrid neutrons stars is the strange star that stars with strangelet collision. Stranglets are types of meta-stable particles that contain large numbers of quarks and strange quarks are believed to be in the majority.
These stranglets can bcan produce strange stars that are overdese(8 km or less) and have a core of quark matter.It is possible the strange quark matter is in  more stable than nuclear matter and is favored at some point in colapse.In this phase it is called color superconductiviy.See article on strange hybrids https://arxiv.org/pdf/astro-ph/0407155.pdf.
Lattice Quantum chromodynamical simulations  do not provide a conclusive guide yet as to the exact transition  or behavior. So the state is still much debated but the color superconductivity state can be analogous to a type two superconductor with channels for the states of energy of the quarks in this phase.
So stellar cores that pass this phase can be  hybrid stars.If stellar black hole of 3.8- 8 solar masses and has  accretion disk.Along with it being Kerr-type it could produce a quark star in said disk.Kerr types may have sufficient rotational energy that it can produce strange quarks in majority in the disk(150 Mev per femtometer) and hence meta-stable quark matter.
See article on it https://arxiv.org/pdf/0908.2672.pdf.
