Common signals in public channels of cellular systems are usually transmitted omnidirectionally from the base station (BS). In recent years, both discrete and consecutive omnidirectional space-time block codings (STBC) have been proposed for massive multiple-input multiple-output (MIMO) systems with a uniform linear array (ULA) configuration to ensure cell-wide coverage. In these systems, constant received signal power at discrete angles, or constant received signal sum power in a few consecutive time slots at any angle is achieved. In addition, equal-power transmission per antenna and full spatial diversity can be achieved as well. In this letter, by utilizing the property of orthogonal complementary codes (OCCs), a new consecutive omnidirectional quasi-orthogonal STBC (QOSTBC) design is proposed, in which constant received sum power at any angle can be realized with equal-power transmission per antenna through one STBC transmission, and a higher diversity order of 4 can be achieved. In addition, the proposed design can be further extended to the uniform planar array (UPA) configuration with the two-dimensional OCCs.