We demonstrate a quasi-periodic structure exhibiting multiple photonic band gaps (PBGs) based on sub-micron-period poled lithium niobate (LN). The structure consists of two building blocks, each containing a pair of antiparallel poled domains, arranged as a Fibonacci sequence. The gap wavelengths are analyzed with the Fibonacci sequence parameters such as the quasiperiodic indices and the average lattice parameter. The transmission properties are investigated by a traditional 4\times4 matrix method. It has also been proved that the gap depth can be tuned by the lengths of poled domains.