With current advances in gravitational wave (GW) detection made by the worldwide LIGO-Virgo-KAGRA (LVK) network of detectors, ever-more sensitive tests of gravity in the strong-field regime are now possible. This enables one to test gauge theories beyond Einstein-Hilbert action, such as Weyl gauge theories of gravity. The only anomaly-free (quantum) gauge theory of a space-time symmetry beyond Poincar'e is based on Weyl gauge group (of dilatations and Poincar'e symmetry) with Weyl conformal geometry as its natural underlying geometry. This gauge theory has spontaneous breaking of Weyl gauge symmetry to Einstein-Hilbert and Proca actions, plus a positive cosmological constant, with the mass of the Weyl-Proca field a free parameter of the theory. We investigate the GW polarisation modes of Weyl (quadratic) gauge theory of gravity in Weyl geometry and compare our findings to the most recent experimental data. We show how the geodesic deviation equation from Riemannian geometry translates to Weyl geometry, and explain why it is crucial to perform the analysis around de Sitter background, which is the correct low-energy limit of Weyl quadratic gravity, to not alter the GW content, and then compute the polarisation modes. In addition to the two transverse-traceless tensor modes predicted by Einstein-Hilbert action, we find two additional vector modes induced by the transverse fluctuations of the Weyl-Proca gauge field. If this field is ultralight ($m_{\omega }\sim {10}^{-13}$ eV), it can be detected by LVK, in which case these vector modes can be important evidence for Weyl gauge symmetry.