Diagnosis
That's carbon "fines" or loose media from inside the new filter elements. It's not harmful (it's just activated carbon), but it's understandably annoying and means the filters weren't primed thoroughly enough.
How to Fix It
You need to re-prime the filters, aggressively. Remove them. Fully submerge each filter in a bowl or bucket of water. Vigorously shake and pump them up and down underwater for a good minute to force all the air out and flush any loose particles. You'll see a cloud of black dust come out. Keep doing it until the water runs clear. Then reinstall.
If That Didn't Work
If specks persist after several priming cycles and flushing a few upper chambers of water through, contact Berkey customer support. There's a small chance a filter element is defective, but 99% of the time it's just an incomplete initial priming.
Heads Up
This is why the priming instructions are a pain but important. That first batch of filtered water after installing new filters should often be discarded or used for plants, not drinking.