Nitrogen oxides play a significant role in various biomedical conditions, including respiratory disorders, asthma, and cardiovascular problems, underscoring the urgent need for sensitive and selective devices in biomedical applications. This study offers a comprehensive analysis of the sensitivity of β-tellurene doped with 2.22 % tungsten to nitrogen oxides (NO, NO2, and N2O). Site-specific doping of tellurene with tungsten reduces the band gap and introduces magnetization in β-tellurene. The strong adsorption energies observed for NO, NO2, and N2O at site A (-2.45 eV, -2.39 eV, and -2.80 eV, respectively) suggest that W-doped β-Te monolayers are promising candidates for gas storage for these compounds. Conversely, weaker adsorption energies for the same gases at site B (-0.74 eV, -1.74 eV, and -0.09 eV) highlights the importance of doping location. The adsorption energy values at site B indicate that W-doped β-Te monolayers have potential as sensing materials for NO and as adsorbents for NO2 gas. Conversely, the weak adsorption energy for N2O at the B site demonstrates its non-interacting behaviour with the W-doped β-Te monolayer. Additionally, the negligible change in electronic properties and minimal charge transfer suggest that this configuration is unsuitable for N2O storage and sensing. The spin-resolved current-voltage characteristics of doped tellurene reveal distinct behaviors influenced by gas molecule adsorption. Overall, these findings underscore the potential of W-doped tellurene as a site-specific material for the adsorption and sensing of targeted gases.