Bragg resonances have been observed in traditional quasiperiodic waveguides for years. To enhance Bragg resonances and form Bragg gaps with high attenuation, a large number of units have been included. Here, a Fibonacci quasiperiodic acoustic waveguide based on an intense resonance mechanism is proposed to realize non‐Bragg resonances. In a waveguide with fewer units, non‐Bragg resonances can still be excited and create the related non‐Bragg gaps with high attenuation beyond the Bragg ones. Furthermore, an interface state arising in non‐Bragg gaps is experimentally observed by fabricating the mirror‐symmetry structure of Fibonacci quasi‐periodic waveguides. The field distribution of interface states exhibits a significant feature of sound energy, which is localized at a quarter wavelength from the interface of the mirror structures on both sides. Unlike the interface state in Bragg gaps (the localized sound energy distributes along the longitudinal direction), the interface state in non‐Bragg gaps displays localization along the longitudinal and radial directions, and the field distribution is like a dipole, which can be used to optimize traditional quasiperiodic waveguide structures and pave the way for various functional devices, which include multimode filters, sensors, couplers, and so on.