Is $\{\langle G,x\rangle \mid x\in L(G)\}$ context-free?
If your language were context-free, then by intersecting it with a regular language, we would deduce that the following language would be context-free:
$$
\{ \langle G,x \rangle \mid \text{$x \in L(G)$ and $G$ contains a single production $S \to w$} \}
$$
However, this is just the celebrated language $\{ ww : w \in \Sigma^* \}$ in disguise, which is known not to be context-free. It follows that your language (using any reasonable encoding) is also not context-free.