The rapid proliferation of generative AI, especially large language models, has led to their integration into a variety of applications. A key phenomenon known as weak-to-strong generalization - where a strong model trained on a weak model's outputs surpasses the weak model in task performance - has gained significant attention. Yet, whether critical trustworthiness properties such as robustness, fairness, and privacy can generalize similarly remains an open question. In this work, we study this question by examining if a stronger model can inherit trustworthiness properties when fine-tuned on a weaker model's outputs, a process we term weak-to-strong trustworthiness generalization. To address this, we introduce two foundational training strategies: 1) Weak Trustworthiness Finetuning (Weak TFT), which leverages trustworthiness regularization during the fine-tuning of the weak model, and 2) Weak and Weak-to-Strong Trustworthiness Finetuning (Weak+WTS TFT), which extends regularization to both weak and strong models. Our experimental evaluation on real-world datasets reveals that while some trustworthiness properties, such as fairness, adversarial, and OOD robustness, show significant improvement in transfer when both models were regularized, others like privacy do not exhibit signs of weak-to-strong trustworthiness. As the first study to explore trustworthiness generalization via weak-to-strong generalization, our work provides valuable insights into the potential and limitations of weak-to-strong generalization.