The accelerated progress in quantum computing has enabled a new form of machine intelligence that runs on quantum hardware, which holds great promise for more powerful computational models in various learning tasks. An emergent application of Quantum Machine Intelligence (QMI) is Quantum Natural Language Processing (QNLP). This paper proposes a multi-dimensional, finite automaton model for quantum word embedding (QWE) via the Galois field. We demonstrated the model to three applications: (1) English vocabulary, (2) amino acid-based genetic codes, and (3) DNA-based genetic codes. The numerical results obtained from the proposed algorithm for the English vocabulary indicate that it produces more representative word features than Word2Vec based on the word distance metric. Second, the proposed algorithm is also utilized to model RNA-Protein interaction based on the latent distance of a given molecule, which is demonstrated on three large datasets, namely RPI369, RPI1807, and RPI2241. Finally, two embedding techniques for DNA-based genetic codes are proposed in this work: Two-state Lackadaisical Encoding (TCE) and Topological-Cyclic Encoding (TLE). These techniques enable extracting relevant features for the efficacy score of gRNAs used in the CRISPR-Cas 9 system, demonstrated on 15 datasets, compared to 12 mathematical features. We make our implementation available at https://github.com/namnguyen0510/Quantum-Embedding-of-Word/tree/main.