This paper introduces the Poisson-New Quadratic-Exponential Distribution (PNQED), a discrete compound probability distribution with a single parameter. It is derived by mixing the Poisson distribution with the New Quadratic-Exponential Distribution (NQED). The PNQED is proposed as a better alternative to existing distributions such as the Poisson-Lindley distribution (PLD), Poisson-Mishra distribution (PMD), and Poisson-Modified Mishra distribution (PMMD) for modeling over-dispersed count data. The probability mass function (pmf) of PNQED is derived and its important characteristics, including the probability generating function (pgf) and moment generating function (mgf), are presented. The first four moments about the origin are calculated, showing that the mean of PNQED decreases as the parameter increases. The paper also discusses methods of estimation and presents graphical results for the pmf and mean of PNQED. The PNQED is found to be a better fit for over-dispersed count data compared to existing distributions. The results show that the PNQED has a more flexible shape and can better model real-world data. The paper concludes that the PNQED is a useful distribution for statistical modeling of over-dispersed count data.This paper introduces the Poisson-New Quadratic-Exponential Distribution (PNQED), a discrete compound probability distribution with a single parameter. It is derived by mixing the Poisson distribution with the New Quadratic-Exponential Distribution (NQED). The PNQED is proposed as a better alternative to existing distributions such as the Poisson-Lindley distribution (PLD), Poisson-Mishra distribution (PMD), and Poisson-Modified Mishra distribution (PMMD) for modeling over-dispersed count data. The probability mass function (pmf) of PNQED is derived and its important characteristics, including the probability generating function (pgf) and moment generating function (mgf), are presented. The first four moments about the origin are calculated, showing that the mean of PNQED decreases as the parameter increases. The paper also discusses methods of estimation and presents graphical results for the pmf and mean of PNQED. The PNQED is found to be a better fit for over-dispersed count data compared to existing distributions. The results show that the PNQED has a more flexible shape and can better model real-world data. The paper concludes that the PNQED is a useful distribution for statistical modeling of over-dispersed count data.