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In this paper, we numerically study Turing patterns by the Finsler geometry (FG) modeling technique on thermally fluctuating triangular lattices, which are often used for modeling cell membranes or lipid membranes, focusing on the origin of diffusion anisotropy. The FG modeling prescription allows us to assume direction-dependent diffusion described by Laplacian. To implement such diffusion anisotropy in the FG modeling, we need an internal degree of freedom (IDF), which depends on direction and position and is controlled by some external forces or stimuli. For such a direction-dependent IDF, we use velocity directions corresponding to thermal fluctuations of the lattice vertices. We find that anisotropic Turing patterns emerge in the direction along which vertices fluctuate. In the simulations, direction-dependent diffusion coefficients are unnecessary for input, and instead, the IDF aligns the direction of vertex fluctuation along a direction implemented by external stimuli. Our results and techniques provide insight into the origin of diffusion anisotropy connected to Turing patterns.