The LAPACK routines GEQRT2 and GEQRT3 can be used to compute the QR decomposition of a matrix of size m×n as well as the storage-efficient representation of the orthogonal factor . A GPU-accelerated algorithm is presented that expands a blocked CPU-GPU hybrid QR decomposition to compute the triangular matrix T. The storage-efficient representation is used in particular to access blocks of the matrix Q without having to generate all of it. The algorithm is presented in two variants using one and two GPUs, respectively. To avoid redundant computations or communication between devices, a scheme is developed to additionally compute during the iteration. As a result the algorithm outperforms the standard LAPACK routine by a factor of 3 for square matrices on a single GPU and by a factor of 5 on two GPUs.