This work studies multiuser detection for one-bit massive multiple-input multiple-output (MIMO) systems in order to diminish the power consumption at the base station (BS). A low-complexity near-maximum-likelihood (nML) multiuser detection algorithm is designed, assuming that each BS antenna port is connected with a pair of single-bit resolution analog-to-digital converters (ADCs) and each user equipment (UE) transmits symbols from a quadrature amplitude modulation (QAM) constellation. First, a novel convex program is formulated as a convex surrogate of the ML detector and subsequently solved through an accelerated first-order method. Then, the solution of the convex optimization problem is harnessed to solve a refined combinatorial problem with reduced search space, requiring non-exponential complexity on the number of the UEs. Judicious simulation study corroborates the efficacy of the resulting two-phase detection algorithm. The proposed two-phase algorithm can achieve symbol error rate (SER) performance close to the ML detector, with significantly reduced computation cost compared to the nML detection schemes in prior art.