(Too long for a comment.)
The sequence of integers such that $x^2-py^2=-1$ is solvable is given by,
$$p = 1, 2, 5, 10, 13, 17, 26, 29, 37, 41, 50, 53, 58, 61, 65, 73, 74, 82, 85, 89, 97, 101,\dots$$
which is A031396, while that of $x^2-2q y^2 = -1$ is (if I did my code right),
$$q = 1, 5, 13, 25, 29, 37, 41, 53, 61, 65, 85, 101, \dots$$
and is not yet in the OEIS.
Point 1: The sequence $p$ does not contain non-square $q$ as a subset.
With limited data, it seems to be the case. But the first missing value is $q=221$, since $x^2-221y^2=-1$ is not solvable, while $x^2-2\cdot221y^2=-1$ is.
Point 2: There is an infinite number of intersections between $p$ and $q$.
Proof: We use the identities,
$$m^2-(m^2+1)\cdot 1^2 = -1$$
$$(2n+1)^2-2\cdot(2n^2+2n+1)\cdot 1^2 = -1$$
Equate,
$$m^2+1 = 2n^2+2n+1$$
and turns out to be a well-known Pell equation in disguise,
$$(2n+1)^2-2m^2 = 1$$
$$u^2-2v^2=1$$
with solutions $(u,v) = (3,2),\,(17,12),\,(99,70),\dots$ and proves there is an infinite number of $d$ such that
$$x^2-dy^2 = -1$$
$$x^2-2dy^2 = -1$$
is both solvable.
P.S. However, to characterize all $d$ seems to be difficult.