Ternary Quadratic Forms and Norms edited by Olga Taussky (1982). Pages 5-30 is William Plesken, Automorphs of Ternary Quadratic Forms. The word automorph is one of the traditional terms for what would now be called a member of the (integer) automorphism group of the form, or rotation group, or orthogonal group.
Since $xz - y^2$ has the mixed coefficient $1,$ we need to double it to get an integral matrix, and this is also the Hessian matrix,
$$ H = \left( \begin{array}{rrr} 0 & 0 & 1 \\ 0 & -2 & 0 \\ 1 & 0 & 0 \end{array} \right) $$
This becomes annoying for diagonal forms, where we still double the diagonal entries: $$ G = \left( \begin{array}{rrr} 38 & 0 & 0 \\ 0 & 10 & 0 \\ 0 & 0 & -2 \end{array} \right) $$
In this direction, take $$ N = \left( \begin{array}{rrr} 38 & 30 & 16 \\ 19 & -25 & -7 \\ -38 & 5 & 9 \end{array} \right) $$ for one of infinitely many solutions to $$ N^T H N = -95 G $$
give me a few more minutes...
In the other direction, take $$ M = \left( \begin{array}{rrr} 2 & 2 & -2 \\ 3 & -8 & -1 \\ 11 & -4 & 9 \end{array} \right) $$ then $$ M^T G M = -380 H $$