This paper proposes a novel second-order cone programming (SOCP) relaxation for a quadratic program with one quadratic constraint and several linear constraints (QCQP) that arises in various real-life fields. This new SOCP relaxation fully exploits the simultaneous matrix diagonalization technique which has become an attractive tool in the area of quadratic programming in the literature. We first demonstrate that the new SOCP relaxation is as tight as the semidefinite programming (SDP) relaxation for the QCQP when the objective matrix and constraint matrix are simultaneously diagonalizable. We further derive a spatial branch-and-bound algorithm based on the new SOCP relaxation in order to obtain the global optimal solution. Extensive numerical experiments are conducted between the new SOCP relaxation-based branch-and-bound algorithm and the SDP relaxation-based branch-and-bound algorithm. The computational results illustrate that the new SOCP relaxation achieves a good balance between the bound quality and computational efficiency and thus leads to a high-efficiency global algorithm.