Flow-induced noise is a common problem in practical engineering. The classical acoustic analogy model is insufficient to evaluate the characteristic distribution of the acoustic field using only acoustic pressure as a reference. This paper starts from a four-dimensional linear wave equation with sound pressure and sound velocity vectors as variables, by choosing the Kirchhoff surface to enclose a nonlinear acoustic source and combining with the convective Green"s function, the four-dimensional acoustic frequency-domain integral equation for a uniformly moving medium is developed. Numerical prediction studies were conducted for stationary, rotating monopole and dipole sources. The results indicate that the distributions of the sound pressure and acoustic velocity obtained by the present proposed method are in good agreement with the analytical solutions. In contrast to the stationary flow case, the acoustic field distribution of the stationary point source in the uniform flow exhibits a convection effect. On the other hand, the acoustic field distribution of the rotating point source exhibits a strong Doppler effect and convection effect due to the joint influence of the uniform flow, the self-excitation frequency, harmonic order, and rotational frequency of the point source. The refined study of aerodynamic vector noise with uniform flow in this paper can provide technical support for the assessment of acoustic energy and the prediction of transmission paths. Additionally, it provides a theoretical basis for noise reduction studies.