Well, the idea behind dark matter is essentially one of dark mass: we can see that the universe (according to our potentially faulty cosmological models) or at least the Milky Way (according to much simpler and probably correct models) has more gravity than the visible matter can account for. So dark matter is "anything that isn't visible to us today and has mass".
By that definition, any star system we cannot see today would be "dark matter". We can see stars, so none of those qualify as dark matter as such - but perhaps the star has invisible (that is, very dark) planets circling it in surprising abundance, explaining some of the extra gravity? That'd be a perfectly possible "dark star system", although it wouldn't explain more than a tiny fraction of the extra gravity.
OTOH, even objects that we can see, such as nebulae, may have something inside them that we cannot see or recognize as the source of gravity. In Star Trek, nebulae tend to be very dense, bright and often surprisingly compact in size - all hints that they have more gravity than they ought to. So perhaps there's dark matter there in addition to the known gases, possibly even enough of it to explain a significant fraction of the "missing" gravity of our galaxy. Whether that matter would accrue inside stars and planets when those are born out of the nebulae is unknown - but again, Trek is famous for dense asteroid belts and small planetoids that have surprisingly much surface gravity to facilitate our heroes walking on them, or breathing an atmosphere that clings to that surface. Perhaps there's dark matter there, too?
Trek also has mastered the artificial creation or negation of gravity. Perhaps the dark matter of the universe is actually exhausts and byproducts from all the advanced civilizations that process gravity, and have done so for billions of years? Wouldn't explain broader cosmology, as dark matter supposedly has existed on universe-wide scale long before technology could plausibly have emerged. Could explain the galaxies, though.
But as dark matter is just matter we haven't seen so far, with mass/gravity, it's probably perfectly possible to have life-as-we-know-it built out of matter that contains a smidgen of dark matter, the only effect being a bit of extra mass. That's more or less the definition: dark matter doesn't have any effects on anything except through its mass - that's why it's dark/invisible to us.
...Of course, all sorts of dark matter could interact with the rest of the universe through ways we don't recognize today, such as subspace radiation or warp attraction or nadion interaction or whatever futuristic Trek phenomena we have. Star systems containing significant amounts of that stuff would then have all sorts of weird effects on our heroes, or at least on our heroes' technology...
Timo Saloniemi