<p>Complex q-Rung orthopair fuzzy sets (Cq-ROFSs) have emerged as a powerful tool for representing uncertainty and imprecise information. As a generalized form of complex intuitionistic fuzzy sets (CIFSs) and complex pythagorean fuzzy sets (CPFSs), they extend flexibility by allowing each element to possess multiple pairs of truth and falsity membership degrees, encompassing both amplitude and phase components. The proposed method employs Hamacher operator-based sum and product within the Cq-rungOFSs framework to enhance the accuracy and flexibility of aggregation, effectively capturing complex relationships between amplitude and phase membership degrees. This research introduces the concept of a complex q-rung orthopair fuzzy Hamacher graph structure (Cq-rungOFHGS), which integrates graph structure with Cq-rungOFS under the Hamacher Operator. Moreover, building upon the previously proposed Cq-rungOFHGS framework, this study introduces a novel method for calculating the first Zagreb index energy, first Zagreb index RandiĆ energy, and first Zagreb Harmonic energy. Further, the proposed energies for the upper and lower bounds are analyzed using relevant examples. Moreover, the proposed method is applied to assess the impact of nutrition and diet-related health issues in India, providing valuable insights into the most affected regions. Furthermore, an algorithm is developed for the proposed method to facilitate its application in real-world scenarios.</p>