Is there example of a smooth, projective, complex $3$-fold $X$, having $b_{2}(X)=2$ a Mori extremal contraction $\phi: X \rightarrow X'$ which contracts a smooth quadric surface $Q \subset X$?
It doesn't matter which of the two possible types it is, i.e. if $\phi(Q)$ is an ODP or $\{xy-z^2-t^3=0\} \subset \mathbb{C}^{4}$.