Experimental results of Janus particles moving in a viscoelastic fluid with a polymeric solute show an increase in the rotational diffusion with the propulsion velocity (Gomez-Solano et al. 2016), and sustained rotation beyond a certain value of the velocity (Narinder et al. 2018). A mechanism that can induce the spontaneous rotation of active colloids is an instability resulting from a fore-aft asymmetric interaction between the particle and the polymers, leading to a torque (De Corato et al., 2021). This mechanism is based on an advective instability that requires repulsive and attractive interactions between the solute and the surface of the particle along the direction of motion, combined with an active mechanism that generates a dipolar distribution of the polymer concentration. In this work, we apply a simulation approach based on a finite-element method to investigate the effects of thermal fluctuations on the motion of an active Janus particle in a polymer solution. We model the evolution of the polymer concentration using an advection-diffusion equation, which includes thermal fluctuations. The results show that driving a Janus particle out of equilibrium in a polymer solution has a profound impact on its Brownian dynamics and can drastically enhance the rotational diffusion.