Generative artificial intelligence methods are employed for the first time to construct a surrogate model for plasma turbulence that enables long-time transport simulations. The proposed GAIT (Generative Artificial Intelligence Turbulence) model is based on the coupling of a convolutional variational autoencoder that encodes precomputed turbulence data into a reduced latent space, and a recurrent neural network and decoder that generate new turbulence states 400 times faster than the direct numerical integration. The model is applied to the Hasegawa-Wakatani (HW) plasma turbulence model, which is closely related to the quasigeostrophic model used in geophysical fluid dynamics. Very good agreement is found between the GAIT and the HW models in the spatiotemporal Fourier and Proper Orthogonal Decomposition spectra, and the flow topology characterized by the Okubo-Weiss decomposition. The GAIT model also reproduces Lagrangian transport including the probability distribution function of particle displacements and the effective turbulent diffusivity.