The q-composite key predistribution scheme  is used prevalently for secure communications in large-scale wireless sensor networks (WSNs). Prior work  explores secure connectivity in WSNs employing the q-composite scheme for q = 1 with unreliable communication links modeled as independent on/off channels. In this paper, we investigate secure connectivity in WSNs operating under the q-composite scheme for general q and under the on/off channel model. We present conditions on how to scale the model parameters so that the network is securely connected with high probability when the number of sensors becomes large. The results are given in the form of zero-one laws. Numerical experiments confirm the validity of our analytical results.