Set of locations where the Hilbert symbol is not equal to $1$

I don't know of a common term for this set, but calling it a Hilbert set or something similar would be reasonable. Let me propose something only slightly fancier below.

Note that $(a,b)_v=-1$ if and only if the quaternion algebra $\langle a,b\rangle_\mathbb{Q}$ (among hundreds of other notations) is ramified at $v$, i.e., if $v$ divides its discriminant. So if you were in a setting where, say, algebraic geometry language was convenient, you might call this set the "Hilbert support of $\langle a,b\rangle$", or maybe reference the Hilbert radical if it were more convenient to refer to the product of such primes (which is occasionally useful).

Edit: I see that SAGE calls the Hilbert conductor what I call the Hilbert radical. That works too.