How is quark produced?

Answer 1

Current scientific theory places the production (creation) of naturally occurring quarks prior to the end of the inflationary period; later during the "hadron epoch" after the Big Bang when the universe had cooled sufficiently, the familiar baryons, composed of quarks, could form - such as neutrons and protons - and were able to coalesce from the quark-gluon plasma.

In the laboratory setting it is possible to create quarks in high energy collisions; the energy of the collision needs to be adequate to produce them - i.e., it would have to be at least equal to their mass-energy as described by the matter-energy equivalence equation (E=mc^2). The greater the mass, the more energy is required. Because it's not completely predictable, the production of a quark is somewhat random, other particles may be produced in any given collision, and it is more of a statistical phenomenon than a deterministic method.