  We discuss the space of sections and certain bisections on a quadric surfaces
bundle $X$ over a smooth curve. The Abel-Jacobi from these spaces to the
intermediate Jacobian will be shown to be dominant with rationally connected
fibers. As an application, we prove that the integral Hodge conjecture holds
for degree four integral Hodge classes of fourfolds fibered by quadric bundles
over a smooth curve. This gives an alternative proof of a result of
Colliot-Th{\'e}l{\`e}ne and Voisin.
