We present an inexact spectral bundle method for solving convex quadratic symmetric cone programming (CQSCP). This method is a first-order method, hence requires much less computational cost in each iteration than second-order approaches such as interior-point methods. In each iteration of our method, we compute a largest eigenvalue inexactly, and solve a small convex quadratic symmetric cone program as a subproblem. We give a proof of the global convergence of this method using techniques from the analysis of the standard bundle method, and investigate Lipschitzian error bounds for the CQSCP problem under some mild assumptions. Finally, we describe an application of our proposed algorithm to convex quadratic semidefinite programming problems. Numerical experiments with matrices of order up to 2000 are performed, and the computational results establish the effectiveness of this method.