While conventional Z-scan techniques using Gaussian beams have been extensively studied for nonlinear absorption measurements, structured beams like Mathieu beams offer unique advantages through their tunable intensity distributions and focusing characteristics, yet their analytical modeling for nonlinear absorption remains largely unexplored. Understanding how different beam parameters influence nonlinear optical interactions is crucial for optimizing experimental configurations and developing advanced nonlinear optical devices. This paper presents a comprehensive analytical model for nonlinear absorption in Z-scan measurements using Mathieu beams. Under the weak nonlinearity approximation, we derive closed-form expressions for transmitted optical intensity, power, and normalized transmittance, incorporating both linear and nonlinear absorption effects. Through extensive numerical simulations, we systematically investigate the influence of four key beam parameters: beam order, ellipticity parameter, Rayleigh range, and radial cutoff parameter on nonlinear absorption characteristics. The results demonstrate that higher-order Mathieu beams produce progressively stronger nonlinear absorption effects while simultaneously narrowing the Full Width at Half Maximum of Z-scan curves, indicating enhanced focusing characteristics. Increasing beam ellipticity enhances absorption strength while narrowing the interaction region, reflecting the direct relationship between ellipticity and intensity concentration. The Rayleigh range analysis confirms the fundamental trade-off between beam confinement and interaction length, where tighter focusing enhances peak intensities at the expense of interaction volume. The radial cutoff parameter investigation emphasizes the importance of spatial sampling in Z-scan measurements. These findings establish a comprehensive framework for understanding and optimizing nonlinear optical interactions with Mathieu beams, providing valuable insights for designing nonlinear optical devices and optimizing Z-scan experimental configurations, with unprecedented flexibility for applications in optical limiting, beam shaping, and nonlinear optical switching systems.