Positron emission tomography (PET) is a widely used molecular imaging technique with many clinical applications. To obtain high quality images, the amount of injected radiotracer in current protocols leads to the risk of radiation exposure in scanned subjects. Recently, deep learning has been successfully used to enhance the quality of low-dose PET images. Extending this to "zero-dose," i.e., predicting PET images based solely on data from other imaging modalities such as multimodal MRI, is significantly more challenging but also much more impactful. In this work, we propose a attention-based framework that uses multi-contrast MRI to reconstruct PET images using the most commonly-used radiotracer, 18F-fluorodeoxyglucose (FDG), a marker of metabolism. We also introduce an input dropout training strategy to handle possible missing MRI contrasts. We evaluate our methods on a dataset of patients with brain tumors, showing the ability to create realistic and clinically-meaningful FDG brain PET images with low errors compared with full-dose ground truth PET images.