Due to their flexibility, battery powered or energy-harvesting wireless networks are deployed in diverse applications. Securing data transmissions between wireless devices is of critical importance in order to avoid privacy-sensitive user data leakage. In this paper, we focus on the scenario where some nodes are curious (but not malicious) and try to identify the initial states of one (or more) other nodes, while some other nodes aim to preserve the privacy of their initial states from these curious nodes. We present a privacy-preserving finite transmission event-triggered quantized average consensus algorithm. Its operation is suitable for battery-powered or energy-harvesting wireless network since it guarantees (i) efficient (quantized) communication, and (ii) transmission ceasing (which allows preservation of available energy). Furthermore, we present topological conditions under which the proposed algorithm allows nodes to preserve their privacy. We conclude with a comparison of our algorithm against other algorithms in the existing literature.