I'm pretty sure the simple answer is that they (quarks) don't interact with the Higgs field, but instead they (quarks) inherit their mass from the binding energy of the gluon field (i.e. they get the mass via totally separate mechanism than from the Higgs); or at least thats where most of it comes from.

Try reading up on QED (Quantum chromodynamics) and/ or check out this related video from Veritasium if your interested in learning more:  https://www.youtube.com/watch?v=Ztc6QPNUqls

I'm pretty sure the simple answer is that they (quarks) do get their intrinsic mass via interactions with the Higgs field (via exchange of "Weak Hypercharge").

However, given that quarks cannot exist on their own, composite particles made of quarks (protons, neutrons, etc.) will end up getting far more of their intrinsic mass from the binding energy of the gluon field (i.e. particles made of quarks end up getting most of their mass via totally separate mechanism than the Higgs; via the exchange of gluons and virtual particles).

Try reading up on QED (Quantum chromodynamics) and/ or check out this related video from Veritasium if your interested in learning more on this subject: 

https://www.youtube.com/watch?v=Ztc6QPNUqls

https://youtu.be/FoR3hq5b5yE