Aspect-based sentiment analysis (ABSA) is a pivotal area within natural language processing (NLP) that focuses on extracting fine-grained sentiment information from text. A particularly demanding task within ABSA is Aspect Sentiment Quadruplet Extraction (ASQE), which aims to identify quadruplets comprising aspect terms, their corresponding opinion terms, sentiment polarity, and categories. This level of analysis is crucial for downstream applications such as sentiment monitoring and user experience research, offering nuanced insights into textual data. However, current methodologies fall short of fully leveraging the rich linguistic features of sentences and the semantic information of labels embedded within sentences. To address this gap, this paper introduces a novel framework, the Label-Semantics Enhanced Multi-layer Heterogeneous Graph Convolutional Network (LSEMH-GCN), specifically designed for ASQE. Our model integrates sentence linguistic features and label semantics to construct a graph neural network tailored for this task. It employs a multi-layer graph convolutional network that synergizes various linguistic features, and utilizes Biaffine attention to enrich the label probability distribution for token pairs with semantic label information. Furthermore, our approach introduces a token pair vector concatenation strategy alongside an advanced asymmetric label tagging scheme to enhance quadruplet extraction. Comprehensive evaluations on benchmark datasets reveal that LSEMH-GCN significantly surpasses existing state-of-the-art models, establishing a new benchmark for ASQE. Our model achieves an average F1 score improvement of 15.52% and 12.30% on the Restaurant-ACOS and Laptop-ACOS datasets, respectively.