How were quarks made?

Above a certain temperature, quarks get annihilated as quickly as they form. the universe had to cool off a bit to get below that threshold temperature, so the quarks would persist. Then they acquire their mass at about 10^-12 seconds.

The quarks were thought to be created from the potential energy of the decaying "inflaton" field at the end of the inflationary epoch. Any particles present prior to inflation would have become diluted to essentially zero number density by inflation. Indeed, this was one of the early motivations for the inflation theory - it successfully explained why we don't see exotic objects like magnetic monopoles, because their number density is extraordinarily low post-inflation.

At this point in time the weak force and electromagnetic force were unified and all the particles present (quarks, W, Z, gluons, photons, Higgs) were massless.

It was not until at around $10^{-12}$ s and electroweak symmetry breaking, that particles attained their masses via the Higgs mechanism and became the particles we know in the standard model.

At $10^{-12}$ s, the universe had a temperature roughly equivalent to the electroweak scale of 160 GeV. This energy scale sets roughly the maximum mass of particles that can be created at that time and indeed the top quark has a mass of around 170 GeV.