$$\begin{array}{ll} \text{minimize} & f(x) + g(y)\\ \text{subject to} & xy \ge a\\ & x \ge 0\\ & y \ge 0\end{array}$$
where both $f$ and $g$ are convex quadratic functions and $a > 0$. The feasible region is convex and, since $a > 0$, also LMI-representable
$$\{ (x,y) \in \mathbb R^2 : x \geq 0 \land y \geq 0 \land x y \geq a \} = \left\{ (x,y) \in \mathbb R^2 : \begin{bmatrix} x & \sqrt{a}\\ \sqrt{a} & y\end{bmatrix} \succeq \mathrm O_2 \right\}$$
Hence, the original optimization problem can be rewritten as follows
$$\begin{array}{ll} \text{minimize} & f(x) + g(y)\\ \text{subject to} & \begin{bmatrix} x & \sqrt{a}\\ \sqrt{a} & y\end{bmatrix} \succeq \mathrm O_2\end{array}$$
Introducing optimization variables $s, t \in \mathbb R$, we rewrite the optimization problem in epigraph form
$$\begin{array}{ll} \text{minimize} & s + t\\ \text{subject to} & f(x) \leq s\\ & g(y) \leq t\\ & \begin{bmatrix} x & \sqrt{a}\\ \sqrt{a} & y\end{bmatrix} \succeq \mathrm O_2\end{array}$$
Let $f$ and $g$ be
$$f (x) := f_0 + f_1 x + f_2 x^2 \qquad\qquad\qquad g (y) := g_0 + g_1 y + g_2 y^2$$
where $f_2, g_2 > 0$ (to ensure convexity). Inequality constraints $f(x) \leq s$ and $g(y) \leq t$ can be written in LMI form, as follows
$$\begin{bmatrix} 1 & \sqrt{f_2} \, x\\ \sqrt{f_2} \, x & s - f_0 - f_1 x\end{bmatrix} \succeq \mathrm O_2$$
$$\begin{bmatrix} 1 & \sqrt{g_2} \, y\\ \sqrt{g_2} \, y & t - g_0 - g_1 y\end{bmatrix} \succeq \mathrm O_2$$
These LMIs introduce inequalities $s - f_0 - f_1 x \geq 0$ and $t - g_0 - g_1 y \geq 0$, which are redundant. Note that lines $s = f_0 + f_1 x$ and $t = g_0 + g_1 y$ are tangent to the graphs of $f$ and $g$, respectively.
Hence, we obtain a semidefinite program (SDP) in variables $x, y, s, t \in \mathbb R$
$$\begin{array}{lc} \\ \text{minimize} & s + t\\\\ \text{subject to} & \begin{bmatrix} 1 & \sqrt{f_2} \, x & & & \\ \sqrt{f_2} \, x & s - f_0 - f_1 x & & & \\ & & 1 & \sqrt{g_2} \, y & \\ & & \sqrt{g_2} \, y & t - g_0 - g_1 y & \\ & & & & x & \sqrt{a}\\ & & & & \sqrt{a} & y\end{bmatrix} \succeq \mathrm O_6\\\\\end{array}$$
which can be solved numerically using any SDP solver.
The problem is quasi-convex in that it is convex for every fixed $a$ and the feasible set decreases for increasing $a$. Hence it can be solved by bisection in $a$
Example code on bisection to explain strategy
https://yalmip.github.io/example/decayrate/
If you use MATLAB/YALMIP, you can do it very easily using built-in functionality (here using mosek as SDP solver)
A = randn(5);A = A*A';
C = randn(5);C = C*C';
D = randn(5);
Y = sdpvar(5);
sdpvar a
b = 1;c = 1;
Model = [a*trace(A*Y) - trace(C*Y) + a*b <= 0, Y >= 0, trace(D*Y) <= c];
bisection(Model,-a,sdpsettings('solver','mosek'))
I belive the problem is ill-posed though (unless $D$ has certain structure relative to $A$ and $C$) which allows $Y$ to grow arbitrarily large
Best Answer
You may decompose the bilinear constraint $a^TQv$ using a LU decomposition of $Q=LU$. Something like $x=L^Ta$ and $y=Uv$. Next, instead of using the bilinear constraint $x^Ty=0$, a (stronger) complementarity constraint is used $-b_i M\leq x_i\leq b_i M$ and $-(1-b_i) M\leq y_i\leq (1-b_i) M$, where $M$ is a large positive number and $b$ is a boolean variable.