Let $q$ be a quadratic form on variables $x_1,...,x_n$ with integer coefficients (not necessarily represented by an integer matrix). Assume also that $q$ is positive-definite and primitive.
Let $A$ be the matrix representing $q$. We define $\text{adj}(q)$ as the quadratic form represented by the adjoint matrix $\text{adj}(A)$.
If $n=1$, then clearly $A=(1)$ and $\text{adj}(A)=(1)$, so $q$ and $\text{adj}(q)$ are the same form.
For $n=2$, if $A=\left(\begin{matrix}a & b\\b & c\end{matrix}\right)$ then $\text{adj}(A)=\left(\begin{matrix}c & -b\\-b & a\end{matrix}\right)$, which can be written as $\text{adj}(A)=M^tAM$, where $M:=\left(\begin{matrix}0 & 1\\-1 & 0\end{matrix}\right)\in\text{SL}(2,\Bbb{Z})$. This means that $q$ and $\text{adj}(q)$ are equivalent, therefore represent the same integers.
Conclusion: for $n=1,2$ the forms $q$ and $\text{adj}(q)$ represent the same integers.
For $n=3$, this is not necessarily true. For example, take $q=3x_1^2+x_2^2+x_3^2$, whose adjoint is $\text{adj}(q)=x_1^2+3x_2^2+3x_3^2$. Clearly, $q$ represents $2$ but $\text{adj}(q)$ doesn't.
My question is: is there some general property relating representable integers of $q$ and of $\text{adj}(q)$?
Remark: if some additional hypothesis can make things simpler, feel free to use it.