Security is an important issue in wireless sensor networks (WSNs), which are often deployed in hostile environments. The q-composite key predistribution scheme has been recognized as a suitable approach to secure WSNs. Although the q-composite scheme has received much attention in the literature, there is still a lack of rigorous analysis for secure WSNs operating under the q-composite scheme in consideration of the unreliability of links. One main difficulty lies in analyzing the network topology whose links are not independent. Wireless links can be unreliable in practice due to the presence of physical barriers between sensors or because of harsh environmental conditions severely impairing communications. In this paper, we resolve the difficult challenge and investigate k-connectivity in secure WSNs operating under the q-composite scheme with unreliable communication links modeled as independent on/off channels, where k-connectivity ensures connectivity despite the failure of any (k - 1) sensors or links, and connectivity means that any two sensors can find a path in between for secure communication. Specifically, we derive the asymptotically exact probability and a zero-one law for k-connectivity. We further use the theoretical results to provide design guidelines for secure WSNs. Experimental results also confirm the validity of our analytical findings.