In this study, a quasi-periodic layered structure based on a Fibonacci sequence with a central defect layer is investigated, where the surrounding layers are made of metamaterials with negative permittivity and negative permeability. The optical response of the structure is analyzed using the transfer matrix method. The results show that the transmission spectrum exhibits two distinct band gaps: a zero-effective-phase band gap that is independent of the incident angle and a Bragg band gap that depends on the incident angle. The zero-effective-phase band gap corresponds to a spectral region in periodic or quasi-periodic photonic structures where the overall phase change across the structure is nearly zero. Introducing a defect layer leads to sharp transmission peaks inside this band gap, which represent localized resonant modes. The behavior of these defect modes strongly depends on the polarization of the incident light. For transverse electric (TE) polarization, good angular stability is observed, with the quality factor increasing from 106.76 at normal incidence to 112.26 at 30°, and rising to 233.17 at 60°. In contrast, for transverse magnetic (TM) polarization, the defect modes are strongly influenced by the Brewster effect, resulting in significant spectral broadening at 30° and complete suppression of these modes at 60°. The combination of quasi-periodic order and a defect layer enables precise control of the transmission spectrum and strong field enhancement in the defect region, making the proposed structure suitable for applications in optical sensors, filters, and photonic amplifiers.