Depending on what quanta dark matter truly is made of, the Higgs boson may or may not be able to decay into those quanta. Being further nit-picky, at colliders, where the Higgs boson can be studied, dark matter may or may not be produced, the collider experiment will never be able to tell whether a new particle has a long enough lifetime to be the cosmological dark matter. One thus typically refers to such decays as "Higgs to invisible". This avoids the reference to dark matter and would be the right buzz words to use in an internet search.
Yes, there are many dark matter candidates that the Higgs could decay into, yes, experiments at colliders could be able to detect such events, but current searches have not come up with significant discrepancies.
This paper from the CMS collaboration is a nice example to illustrate these points. Their Figure 1 shows some example Feynman diagrams where the Higgs ("H") decays into invisible particles (invisible in CMS, labeled $\chi$) that might be dark matter:
They go on and search their data for such events. Figure 6 from that paper shows what they find:
The colorful areas in the top portion of that figure are various standard model processes that are expected to contribute to this analysis channel. Black dots are their data, which agrees with the sum of the standard model processes. This is again shown in the middle portion where the ratio of the data to the standard model expectation is everywhere about one. The dashed line is the strongest possible amount of decays that they could attribute to the standard model Higgs to decay to some invisible particles.
This then allows them to calculate the largest possible branching ration of Higgs decaying into new beyond-standard model invisible particles, relative to the decay into standard model invisible particles, as being less than 20%, as they show in their Figure 9: