Brownian motion in a fluid with a temperature gradient is investigated by using Luttinger's method of introducing auxiliary external fields. The Einstein relation for the diffusion coefficient $D=kT/\zeta$ and a similar relation for the thermal diffusion coefficient $D_\mathrm T=\displaystyle n_\sigma kT\frac{1+\eta/kT}{\zeta}$ are obtained ($n_\sigma$ is the density of the Brownian particles, $\zeta$ is the friction constant, and $\eta$ is the heat drag coefficient of the Brownian particles). The expressions obtained are compared with the results of other works on diffusion of Brownian particles in a fluid with a temperature gradient.