The conceptual design study of a high gradient super conducting insertion quadrupole magnet has been carried out in collaboration between KEK and CERN for the Large Hadron Collider (LHC) to be built at CERN. A model magnet design has been optimized to provide a nominal design field gradient of 240 T/m with a bore aperture of 70 mm and an operational field gradient of 225 T/m at 1.9 K under radiation environment with a beam energy deposit of several watts per meter in the superconducting coils. The design and its optimization process are discussed