We solve a question asked by Q. Xu about the order of optimal constants in the vector-valued Littlewood-Paley-Stein inequality. The key point is to establish the L^p-norm equivalence between Lusin area integral and Littlewood-Paley g-function. To this end, we introduce the vector-valued Hardy and BMO spaces associated with certain sectorial operators. The argument relies on the molecular decomposition of the vector-valued Hardy space in addition to Mei's duality technique and Wilson's intrinsic square functions developed in this setting."