We develop a thermodynamically consistent Lindblad approach for quantum stochastic thermodynamics, where each bath dissipator satisfies the local quantum detailed balance condition. We formulate the thermodynamic geometry to determine the optimal protocol in the slow-driving regime. Thermodynamic consistency leads to the joint fluctuation theorem for quantum work and heat, with the corresponding joint statistics obtained from a tilted master equation and generally described by a quasiprobability. We apply this framework to the quantum Brownian motion master equation, refining it to satisfy detailed balance, deriving optimal protocols at different temperatures and friction coefficients, and studying the work statistics. Our approach thus enables the study of nonequilibrium thermodynamics in open quantum systems.