Godzilla and the winged MUTO first battle in Hawaii, specifically around Honolulu. In the 2014 film, Godzilla arrives near the islands as the winged male MUTO attacks, and their brief confrontation is followed by destruction and a tsunami in the Honolulu area.
The sequence is intentionally presented in fragments, including news coverage and distant views, rather than as a long, fully shown monster fight. The winged MUTO is the smaller, flying male of the pair; the larger female later joins him as the creatures move toward their reproductive plan and eventual confrontation with Godzilla.
This encounter is easy to confuse with the film’s later battles. Godzilla and the MUTOs ultimately converge in San Francisco, where the climactic fighting takes place. Hawaii is therefore the correct location for their first battle, while San Francisco is the setting for the extended final showdown. The Honolulu sequence also establishes the film’s emphasis on scale, disaster imagery and humanity struggling to understand the Titans.