Recently, the novel two-dimensional semiconductor material β-TeO₂ has been successfully synthesized in experiments and exhibits excellent optoelectronic properties, attracting growing attention and research interest. Using first-principles calculations based on density functional theory, we systematically investigate the structural stability, electronic properties, and magnetism of monolayer β-TeO₂ containing intrinsic point defects (vacancies, interstitials, and antisite defects) and substitutional defects. Several physical quantities-including defect formation energy, electronic band structure, projected density of states, partial charge density, and magnetic moments-are calculated. Under oxygen-poor conditions, the most readily formed vacancy, interstitial, antisite, and substitutional defects are V_O1, O_i, Te_O2, and F_O, respectively. Under oxygen-rich conditions, the corresponding defects are V_O1, O_i, Te_O2, and Sb_Te. The introduction of vacancy defects (V_O1, V_O2, and V_Te), interstitial defects (Te_i), and antisite defects (O_Te, Te_O1, and Te_O2) does not induce magnetism in monolayer β-TeO₂, and its semiconducting characteristics are preserved. However, these defects introduce multiple spin-degenerate defect states within the bandgap, leading to a significant reduction in bandgap size. In contrast, interstitial oxygen defects (O_i) neither induce magnetism nor produce in-gap defect states β; the system retains its semiconducting character with an essentially unchanged bandgap. Conversely, all substitutional defects (F_O1, F_O2, N_O1, N_O2, I_Te, and Sb_Te) induce magnetism in monolayer β-TeO₂ and generate spin-polarized defect states within the bandgap, transforming the system into a magnetic semiconductor. The corresponding magnetic moments are 0.66μ_B, 0.59μ_B, 0.67μ_B, 0.74μ_B, 0.77μ_B, and 0.75μ_B, respectively. Furthermore, we provide a detailed analysis of the origin and mechanism of the defect states and magnetic moments. This study advances the understanding of defect properties in two-dimensional β-TeO₂ and provides a theoretical foundation for its applications in electronic and spintronic devices.