Is there any one point set of Q which is open?
But there is really no way to answer unless you specify the topology.
Even if you say “with the usual topology”, it is not clear if you mean the set is to be viewed as a subset of some other space or as a space in and of itself.
Suppose we specify the superset is the set of real numbers with the usual topology.
Now, the subset of rational numbers, , we can ask if is an open subset of .
Here is a hint. Every point of an open set is an interior point of the set. But every point of is a boundary point.