We present a novel framework demonstrating how the appropriate encoding scheme, symmetry aware circuit architecture, and measurement protocol provide an alternative route to modeling nonlinear dissipative dynamics in quantum simulations of fluid flows. We focus on the quantum lattice Boltzmann method, where the central difficulty lies in reproducing the nonlinear and non-unitary collision operator. Rather than linearizing the governing equations, we design and learn a shallow surrogate quantum circuit for this operator. The circuit is trained within a quantum circuit learning framework while enforcing physical constraints, including mass and momentum conservation, scale equivariance, and lattice symmetries. We test the approach on the Taylor-Green vortex and lid-driven cavity flows across different Reynolds numbers and provide a theoretical analysis and numerical validation showing that nonlinearity and dissipation emerge in the classical space from the combined effects of data encoding, unitary evolution, and measurement. Finally, we discuss extensions toward multi-timestep evolution without intermediate measurement and outline open challenges.