I would like to know if there exists a smooth complex projective $3$-fold $X$ that admits a fibration $\pi: X\to \mathbb CP^1$ such that all fibers are smooth, $\pi^{-1}(0)$ is the second Hirzebruch surface $F_2$ and for any $x\in \mathbb CP^1$, $x\ne 0$ the fiber $\pi^{-1}(x)$ is a smooth quadric.

I suspect that in case such a three-fold exists, it will not be unique up to an isomorphism. If this is the case, can there be a classification of such three-folds? Is there a "simplest one" among them?

PS. As Piotr points out below, a construction of such a fibration is contained in the answer to this question: A specific degeneration of a rank 2 bundle (one just needs to projectivise the rank $2$ bundle)

I would be grateful to some comments to the second half of the question.