I have a quadratic optimization program with a vector and matrix decision variable. It is of the form
\begin{align} \min_{x \in \mathbb{R}^n, \delta Q \in \mathbb{S}^n} & \frac{1}{2}x^T (Q + \delta Q)^T (Q + \delta Q) x - e^T x\\ \text{ subject to }& x_i \geq 0, \quad 1 \leq i \leq n,\\ &-r \leq \delta Q_{(i,j)} \leq r,\quad 1\leq i,j \leq n, \end{align}
where $r \geq 0$ and $Q \in \mathbb{S}^n$ are constants and $e$ is the vector of ones.
Is there any way I can solve this as an SDP or with an SDP solver?
No, this function is nonconvex, which you can see rather easily by restricting to some sub-space and specific data, for instance $Q=1$ and along the ray $x = t, \delta Q = -2t$.