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As a promising dark matter candidate, the axion particle can naturally solve the strong CP problem of the Standard Model through the U(1) Peccei-Quinn symmetry breaking process. However, detecting this high-energy process poses a significant challenge for colliders. We precisely calculate phase transition gravitational wave signals of this symmetry breaking process in the popular DFSZ axion model. By comparing these signals with the expected sensitivity curves of the Cosmic Explorer and the Einstein Telescope, we demonstrate that Cosmic Explorer can detect this process with a signal-to-noise ratio higher than 8 for the benchmark points. Furthermore, by performing a Fisher analysis, we find that if signals are observed, the bubble wall velocity will be the first phase transition parameter to be determined. Future gravitational wave detectors are expected to provide a new approach for exploring concrete axion models., Comment: References updated, 37 pages, 4 figures, 5 tables, comments are welcome