First, note that the condition that $A$ be positive semidefinite (PSD) doesn't buy you anything. Replacing $A$ by $A+kI$ changes the objective value of any feasible solution by $k$, so if we could solve the given problem when the matrix in the objective is PSD, we could just choose some $k$ large enough to make $A+kI$ PSD, solve the resulting problem with $A+kI$, and subtract $k$ to get the answer to the problem with $A$ instead.
A matrix is called copositive if $x^T A x\geq 0$ for all $x\geq 0$. Checking copositivity of $A$ is the same as checking whether the optimal value of your problem on $-A$ is nonpositive. As shown by Murty and Kabadi ("Some NP-Complete Problems in Quadratic and Nonlinear Programming"), this problem is co-NP-complete. Thus your given problem is NP-hard.
That said, it is a quadratically constrained quadratic program ("QCQP"), which is a well-studied kind of problem, so I would suggest looking into these. For certain classes of them there are well-performing approximation algorithms. In general there are SDP relaxations, although proving performance guarantees is often difficult.
One last thing to note: at least when $A$ is PSD, your formulation is equivalent to one in which the equality constraint is replaced by $\leq$, and then the feasible set will be convex. So you may see the problem in this form.