Quarks are quantum fields inside a nucleus. They interact with gluons. It is in some sense similar to the way in which electrons interact with photons, but this interaction, namely QCD is non-Abelian whereas QED is an Abelian theory. Generators (of color charges) in non-Abelian theories do not commute. The non-Abelian nature leads to three and four gluon couplings and consequently an extremely small interaction range (see Eq. 20-26 in the reference). Nuclear forces, therefore are confined within the nucleus. This phenomenon is quark confinement. QCD is, however, an asymptotically free theory, which means that if we probe inside a nucleus using sufficiently large energies(of proton beams) we find free quarks and their scattering phenomenon, which is quite a mind-boggling thing, rich in phenomenological aspects. Free quarks are not seen in nature, they appear in quark-antiquark bound states in mesons (e.g. pi meson), or, three quarks bound states in baryons (e.g. proton or neutron). Mesons and baryons are collectively called hadrons.
Quarks also couple to photons, but the coupling strength to gluons, expressed by alpha_s ( about 0.125) is about 16 times larger than the interaction strength to photons, expressed by the fine structure constant (about 1/128). This number is also scale dependent, meaning it depends on the energy scale at which one is probing a nucleus. I have quoted it at so called Z-pole , which is about 90 GeV or so.