Numerous methods have been proposed to enhance Keyword Spotting (KWS) in adult speech, but children's speech presents unique challenges for KWS systems due to its distinct acoustic and linguistic characteristics. This paper introduces a zero-shot KWS approach that leverages state-of-the-art self-supervised learning (SSL) models, including Wav2Vec2, HuBERT and Data2Vec. Features are extracted layer-wise from these SSL models and used to train a Kaldi-based DNN KWS system. The WSJCAM0 adult speech dataset was used for training, while the PFSTAR children's speech dataset was used for testing, demonstrating the zero-shot capability of our method. Our approach achieved state-of-the-art results across all keyword sets for children's speech. Notably, the Wav2Vec2 model, particularly layer 22, performed the best, delivering an ATWV score of 0.691, a MTWV score of 0.7003 and probability of false alarm and probability of miss of 0.0164 and 0.0547 respectively, for a set of 30 keywords. Furthermore, age-specific performance evaluation confirmed the system's effectiveness across different age groups of children. To assess the system's robustness against noise, additional experiments were conducted using the best-performing layer of the best-performing Wav2Vec2 model. The results demonstrated a significant improvement over traditional MFCC-based baseline, emphasizing the potential of SSL embeddings even in noisy conditions. To further generalize the KWS framework, the experiments were repeated for an additional CMU dataset. Overall the results highlight the significant contribution of SSL features in enhancing Zero-Shot KWS performance for children's speech, effectively addressing the challenges associated with the distinct characteristics of child speakers.