The integration of generative artificial intelligence (GenAI) into 6G networks promises substantial performance gains while simultaneously exposing novel security vulnerabilities rooted in multimodal data processing and autonomous reasoning. This article presents a unified perspective on cross-domain vulnerabilities that arise across integrated sensing and communication (ISAC), federated learning (FL), digital twins (DTs), diffusion models (DMs), and large telecommunication models (LTMs). We highlight emerging adversarial agents such as compromised DTs and LTMs that can manipulate both the physical and cognitive layers of 6G systems. To address these risks, we propose an adaptive evolutionary defense (AED) concept that continuously co-evolves with attacks through GenAI-driven simulation and feedback, combining physical-layer protection, secure learning pipelines, and cognitive-layer resilience. A case study using an LLM-based port prediction model for fluid-antenna systems demonstrates the susceptibility of GenAI modules to adversarial perturbations and the effectiveness of the proposed defense concept. Finally, we summarize open challenges and future research directions toward building trustworthy, quantum-resilient, and adaptive GenAI-enabled 6G networks.