Much attention has recently been paid to sparsity-aware space-time adaptive processing (STAP) algorithms. The idea of sparsity-aware technology is commonly based on the convex l₁-norm penalty. However, some works investigate the l_q (0 < q < 1) penalty which induces more sparsity owing to its lack of convexity. We herein consider the design of an l_q penalized STAP processor with a generalized sidelobe canceler (GSC) architecture. The l_q cyclic descent (CD) algorithm is utilized with the least squares (LS) design criterion. It is validated through simulations that the l_q penalized STAP processor outperforms the existing l₁-based counterparts in both convergence speed and steady-state performance.