Is there any idea to the following problem?
$$ \begin{array}{rl} \displaystyle\operatorname*{minimize}_{(X,y)\in\mathbb S^{p}\times \mathbb R} & -y\\ \operatorname*{subject to} & F_1 X+X F_1^T+XRX+y Q \preceq 0\\ & X \in \mathbb S^{p}_+\\ & y>0 \end{array} $$
I know it is a convex optimization problem since $R$ is positive semidefinite, but I don't know which kind of problem it belongs to. Thank you for any kind ideas.
As you say, it is convex, so it is currently a convex nonlinear semidefinite program. It is easily rewritten as a linear semidefinite program by performing a Schur complement on the quadratic term, i.e. writing $A - BSS^TB\succeq 0$ as $\left[\begin{array}& A & BS\\S^TB^T &I\end{array}\right]\succeq 0$ (in your case $R = SS^T, B=X, A = -yQ-F_1X-F_1^TX$)
Small detail though that strict inequalities aren't supported in practical theory, you would have to write it as $y\geq \epsilon$ for some small $\epsilon$ in practice. In your case no reason to add any such constraint though. If the largest possible $y$ is non-positive, you know the original problem is infeasible.