What tree drops tiny black seeds/nuts
We can't identify the tree with just the seeds/berries shown. We may be able to do so if you can send photos of the bark, leaf buds, or leaves (either still on the tree, or a clear fallen one.) A photo of the entire tree/shrub and close-ups of branches would help, too.
They resemble the berries of our native black gum tree somewhat. They are very sour and bitter, but edible, technically. Excellent tree.
Whatever these berries are, we have not received questions about problems with them in the past.
Keep in mind that there are many common landscape plants that are toxic, usually mildly so, not fatal, just enough to upset an animal's stomach and teach the animal not to feed on them.
We cannot tell from the trunk photos which species of trees these are; the foliage on the ground is definitely oak, but whether these have blown around from a separate tree or not is not something that can be determined here.
The extension coordinator for your county is Luke Gustafson, who can be reached at firstname.lastname@example.org and 301-934-5404. Perhaps he can take a look at seed samples to ID the tree in question. This assumes, of course, that the seeds were the cause of the dog's reaction.
For the dog's safety, we would recommend consulting a veterinarian about his or her symptoms if you have not already done so. While we realize an ID of the plant matter the dog consumed would be useful, time is of the essence in matters of ingestion of harmful substances and at least the symptoms could be evaluated and treated in the meantime. The vet may also have more experience regarding what specific symptoms may be pointing to with regards to substance exposure.