This paper addresses structured covariance matrix estimation under t-distribution. Covariance matrices frequently reveal a particular structure due to the considered application and taking into account this structure usually improves estimation accuracy. In the framework of robust estimation, the $t$-distribution is particularly suited to describe heavy-tailed observation. In this context, we propose an efficient estimation procedure for covariance matrices with convex structure under t-distribution. Numerical examples for Hermitian Toeplitz structure corroborate the theoretical analysis.