In wireless sensor networks, the $q$-composite key predistribution scheme is a widely recognized way to secure communications. Although connectivity properties of secure sensor networks with the $q$-composite scheme have been studied in the literature, few results address physical transmission constraints since it is challenging to analyze the network connectivity in consideration of both the $q$-composite scheme and transmission constraints together. These transmission constraints reflect real-world implementations of sensor networks in which two sensors have to be within a certain distance from each other to communicate. In this paper, we rigorously derive conditions for connectivity in sensor networks employing the $q$-composite scheme under transmission constraints. Furthermore, we extend the analysis to consider the unreliability of wireless links by modeling each link being independently active with some probability. Our results provide useful guidelines for designing securely and reliably connected sensor networks. We also present numerical experiments to confirm the analytical results.