The Automatic Speaker Verification (ASV) system is vulnerable to fraudulent activities using audio deepfakes, also known as logical-access voice spoofing attacks. These deepfakes pose a concerning threat to voice biometrics due to recent advancements in generative AI and speech synthesis technologies. While several deep learning models for speech synthesis detection have been developed, most of them show poor generalizability, especially when the attacks have different statistical distributions from the ones seen. Therefore, this paper presents Quick-SpoofNet, an approach for detecting both seen and unseen synthetic attacks in the ASV system using one-shot learning and metric learning techniques. By using the effective spectral feature set, the proposed method extracts compact and representative temporal embeddings from the voice samples and utilizes metric learning and triplet loss to assess the similarity index and distinguish different embeddings. The system effectively clusters similar speech embeddings, classifying bona fide speeches as the target class and identifying other clusters as spoofing attacks. The proposed system is evaluated using the ASVspoof 2019 logical access (LA) dataset and tested against unseen deepfake attacks from the ASVspoof 2021 dataset. Additionally, its generalization ability towards unseen bona fide speech is assessed using speech data from the VSDC dataset.