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 $\{xy-z^2-t^3=0\} \subset \mathbb{C}^{4}$.



Mori extremal contraction with small Betti number
As abx mentioned, the simplest example is the blowup of a cubic 3-fold with an ODP. Alternatively, the same variety can be obtained as the blowup of $\mathbb{P}^3$ along a smooth complete intersection of a smooth quadric and a cubic. The strict transform of the quadric then can be contracted to an ODP.


