We study the propagation of transverse acoustic waves in quasi-periodic structures following the Fibonacci sequence made of two blocs A and B where the block A is a metal layer and the bloc B is a piezoelectric layer. The phonon dynamics is described by coupled elastic equations within the static field approximation model. We use the Green's function formalism which enables to get simple analytical expression for the phonon dispersion relation and transmission coefficient. We considered two types of structures: either a given Fibonacci sequence or a periodic repetition of a given sequence called Fibonacci superlattice. The analysis of the transmission spectra of different Fibonacci generations allowed us to conclude that these spectra exhibit self-similarity of order three with a scaling factor F for normal and oblique incidence. Also, we present the results of bulk acoustic waves in Fibonacci superlattices composed of periodic cells where each cell consists of a given sequence. We show the property related to bulk bands such as the fragmentation of the bands as function of the generation number following a power law as well as the existence of stable and transient gaps which are due to the quasi-periodicity.