Generative deep learning techniques are employed in a novel framework for the construction of surrogate models capturing the spatio-temporal dynamics of 2D plasma turbulence. The proposed Generative Artificial Intelligence Turbulence (GAIT) framework enables the acceleration of turbulence simulations for long-time transport studies. GAIT leverages a convolutional variational auto-encoder and a recurrent neural network to generate new turbulence data from existing simulations, extending the time horizon of transport studies with minimal computational cost. The application of the GAIT framework to plasma turbulence using the Hasegawa-Wakatani (HW) model is presented, evaluating its performance via various analyses. Very good agreement is found between the GAIT and the HW models in the spatio-temporal Fourier and Proper Orthogonal Decomposition spectra, the flow topology characterized by the Okubo-Weiss parameter, and the time autocorrelation function of turbulent fluctuations. Excellent agreement has also been obtained in the probability distribution function of particle displacements and in the effective turbulent diffusivity. In-depth analyses of the latent space of turbulent states, choice of hyper-parameters and alternative deep learning models for the time prediction are presented. Our results highlight the potential of AI-based surrogate models to overcome the computational challenges in turbulence simulation, which can be extended to other situations such as geophysical fluid dynamics.