I don't work with time series data much myself. I would imagine you can get at least some transfer learning, since there are patterns that show up across different domains. It looks like there's been a little bit of work done on this: https://arxiv.org/pdf/1811.01533.pdf .

According to them, transfer learning can improve a time series model if you pick the right dataset to transfer from, but they don't seem to be getting the same unbelievably strong transfer results that you'd see on images and text.