No. There is a reciprocity law at play here which places additional restrictions on the discriminant locus.
For example, over $\mathbb{P}^1$ the result is as follows:
Let $D \subset \mathbb{P}^1$ be a reduced divisor over $\mathbb{C}$. Then there exists a non-singular conic bundle $X \to \mathbb{P}^1$ with discriminant locus equal to $D$ if and if $D$ has even degree.
This can be proved using the Faddeev reciprocity law (see e.g. Corollary 6.4.6 in "Gille, Szamuely - Central simple algebras and Galois cohomology").
Further results about conic bundles over surfaces can be found in the paper "Artin, Mumford - Some elementary examples of unirational varieties which are not rational".