I will assume that $X$ is proper. Then the generic fiber is a smooth projective curve of genus $0$ over the function field $K = k(Z)$ but any such curve can be embedded as a conic in $\mathbb{P}^2_K$ using the anticanonical linear series. By Tsen's theorem, the conic has a $K$-point which we can spread out to a $U$-point of $X$ where $U$ is some open subset of $Z$. Finally by propernees, this $U$-point extends to a section over $Z$.

I will assume that $X$ is proper. Then the generic fiber is a smooth projective curve of genus $0$ over the function field $K = k(Z)$ but any such curve can be embedded as a conic in $\mathbb{P}^2_K$ using the anticanonical linear series. By Tsen's theorem, the conic has a $K$-point which we can spread out to a $U$-point of $X$ where $U$ is some open subset of $Z$. Finally by properness, this $U$-point extends to a section over $Z$.