A particle in a uniformly accelerated motion exhibits Brownian random motions around the classical trajectory due to the coupling to the field vacuum fluctuations. Previous works show that the Brownian random motions satisfy the energy equipartition relation. Because this thermal property is understood as the consequence of the Unruh effect, this quantum radiation is termed Unruh radiation. We investigate the properties of Unruh radiation produced by a uniformly accelerating particle undergoing thermal random motions, which originate from the coupling to the vacuum fluctuations of a massless scalar field as well as an electromagnetic field. The energy flux of Unruh radiation is negative and smaller than that of Larmor radiation by one order in a/m, where a is the constant acceleration and m is the mass of the particle. Thus, the Unruh radiation appears to be a suppression of the classical Larmor radiation. The result is consistent with the previous studies on the quantum effect on the Larmor radiation.