Puns, as a linguistic phenomenon, hold significant importance in both humor and language comprehension. While extensive research has been conducted in the realm of pun generation in English, there exists a notable gap in the exploration of pun generation within code-mixed text, particularly in Hindi-English code-mixed text. This study addresses this gap by offering a computational method specifically designed to create puns in Hindi-English code-mixed text. In our investigation, we delve into three distinct methodologies aimed at pun generation utilizing pun-alternate word pairs. Furthermore, this novel dataset, HECoP, comprising of 2000 human-annotated sentences serves as a foundational resource for training diverse pun detection models. Additionally, we developed a structured pun generation pipeline capable of generating puns from a single input word without relying on predefined word pairs. Through rigorous human evaluations, our study demonstrates the efficacy of our proposed models in generating code-mixed puns. The findings presented herein lay a solid groundwork for future endeavours in pun generation and computational humor within diverse linguistic contexts.