Does there exist a simple non-abelian 2-generated group $G$ and two elements $a, b \in G$, such that $\langle \{a, b\} \rangle = G$, $a^2 =1$ and $\forall c, d \in G$ $\langle \{c^{-1}bc, d^{-1}bd \} \rangle \neq G$?



about simple non-abelian 2-generated group
No. Note that $\langle b ,a^{-1}ba \rangle$ is normalized by $b$, and by $a$. Hence 
$\langle b, a^{-1}ba \rangle$ is normalized by $\langle a,b \rangle = G$. Since $G$ is simple non-Abelian, $G = \langle b, a^{-1}ba \rangle .$


