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Solving a 0-1 quadratic program (a combinatorially NP-Hard problem) by converting it into an equivalent 0-1 mixed-integer linear program has been well established in the literature. Several methodologies such as the Reformulation-Linearization Technique (RLT) produce a hierarchy of ever-tightening linear programming (LP) relaxations for constructing the convex hull of the resulting 0-1 mixed-integer linear program. In this research, we add to the body of literature on hierarchical approaches by employing second-order cone programming (SOCP)-based relaxations to improve the solvability of 0-1 QPs. As with other techniques, this hierarchical procedure also generates a sequence of SOCP relaxations that ultimately converges to the convex hull of the underlying 0-1 IP but it offers many computational advantages as the number of constraints required at the kth step of this procedure is significantly lower as compared to existing methodologies, while yet retaining the strength of the underlying lower bound. We prove that this SOCP-based relaxation procedure converges to the convex hull of the 0-1 QP in n steps (where n is the number of binary variables in the problem) and furthermore, preliminary computations on several well-known instances from QPLIB are used to demonstrate the efficacy of the proposed methodology.