Surface acoustic wave (SAW) based devices have been developed in various fields such as for sensing applications. Even more, SAWs are recently studied in interaction with quantum systems, requiring high frequencies and high quality factors (Q factors). Here, we present single-port SAW resonators with resonance frequency above 1 GHz by depositing a layer of piezoelectric material with high phase velocity and designing the fine structure of an interdigital transducer (IDT). In detail, we fabricate SAW resonators on a sputtered aluminum nitride (AlN) layer with IDT finger size of 1 µm by photo-lithography. The devices are characterized by a vector network analyzer, operating at a resonance frequency up to 1.229 GHz with a quality factor Q of 2535 in air. The Q factors are analyzed as equivalent-circuit elements, thus separating external (Q e) and internal (Q i) quality factors. The analysis shows that the external quality factor Q e can more straightforwardly be tuned by design parameters than the internal quality factor Q i. The GHz resonance frequencies and internal quality factors above 1000 can be achieved even without the use of electron beam lithography.