As next-generation mobile networks evolve toward 6G and beyond, intelligent semantic-aware networking across heterogeneous environments becomes paramount for real-time, mission-critical analytics. This work introduces a trust-centric networking framework for the future mobile core, built around a novel, multi-dimensional Quality of Trust (QoT) profile. This profile enables applications to specify dynamic trust requirements, allowing for adaptive and trustworthy service orchestration. The framework is also designed with quantum-ready capabilities to ensure robust data protection in future infrastructures. We validate the core principles of our framework on a testbed emulating an Uncrewed Aerial Vehicle (UAV)-assisted disaster-response mission. Using real-world measurements from classical nodes and analytical models for more secure scenarios, we demonstrate that our trust-aware orchestration preserves the critical latency and bandwidth gains of edge offloading while virtually eliminating processing errors from untrusted nodes.