In this paper, a new multiple-input multiple-output (MIMO) transmission system, which is called space-time quadrature spatial modulation (ST-QSM), and combines quadrature spatial modulation (QSM) with space time block codes (STBCs) in order to achieve second order transmit diversity gain, is proposed. In the proposed ST-QSM scheme, the number of transmit antennas is subdivided into two groups and STBC systems with two, four and eight antennas, depending on the number of antennas in each group, are used to transmit real and imaginary parts of two M-ary quadrature amplitude modulation (M-QAM) symbols. The theoretical error probability analysis of the ST-QSM system is performed and the average bit error probability (ABEP) is obtained. Moreover, it is shown by extensive computer simulations that the ST-QSM system shows better error performance than QSM, space-time coded spatial modulation (STBC-SM) and Alamouti STBC systems.