What sort of observations constrain the QCD interactions of dark matter? Within our current limits of observation, dark matter (DM) is nonluminous i.e., it neither absorbs nor gives off electromagnetic (EM) radiation. This tells that DM is electrically neutral having no EM interaction or puts an upper bound on its electric charge (if any) such that its EM interaction is unobservably tiny!
What sort of observations tell us that DM also cannot have strong QCD interactions? Well, if it has QCD interactions, it will interact with quarks (which are charged), and in turn might cause indirect emission of EM radiation. Right? Is that how observations constrain strong interactions of DM, too?
 A: 
Well, if it has QCD interactions, it will interact with quarks (which are charged), and in turn might cause indirect emission of EM radiation. Right?

Cannot be so, because QCD interactions happen in the dimensions of a fermi,   $1×10^{−15} m$ at most. Very large energies of individual particles are needed for the particles involved in order for QCD effects to dominate, due to the structure of the QCD force.
It is only in the cosmological models that a period where QCD interactions dominate, the quark gluon plasma in the Big Bang model, that such energies can be achieved so that the QCD interactions could dominate. These energies cannot be found in the dark mass needed to explain the galactic rotational curves etc. the energy density is too small:

Dark matter is a form of matter thought to account for approximately 85% of the matter in the universe and about a quarter of its total mass–energy density or about $2.241×10^{−27}$ kg/m3

Discussing within the standard model, energies that allow for QCD direct interactions can be found in neutron stars and supernovas, not in apparent interstellar space, at the end of rotation curves, where dark matter dominates.
Models beyond the standard model propose new particles, but still color neutrality and asymtotic freedom, i.e. no free quarks and gluons, has to be obeyed, so interactions of these particles cannot be strong at the available in space energy level densities.To have effects of  free quarks and gluons needs energies over GeV.
