Alzheimer's disease (AD) is a progressive neurodegenerative disorder leading to cognitive decline. [$^{18}$F]-Fluorodeoxyglucose positron emission tomography ([$^{18}$F]-FDG PET) is used to monitor brain metabolism, aiding in the diagnosis and assessment of AD over time. However, the feasibility of multi-time point [$^{18}$F]-FDG PET scans for diagnosis is limited due to radiation exposure, cost, and patient burden. To address this, we have developed a predictive image-to-image translation (I2I) model to forecast future [$^{18}$F]-FDG PET scans using baseline and year-one data. The proposed model employs a convolutional neural network architecture with long-short term memory and was trained on [$^{18}$F]-FDG PET data from 161 individuals from the Alzheimer's Disease Neuroimaging Initiative. Our I2I network showed high accuracy in predicting year-two [18F]-FDG PET scans, with a mean absolute error of 0.031 and a structural similarity index of 0.961. Furthermore, the model successfully predicted PET scans up to seven years post-baseline. Notably, the predicted [$^{18}$F]-FDG PET signal in an AD-susceptible meta-region was highly accurate for individuals with mild cognitive impairment across years. In contrast, a linear model was sufficient for predicting brain metabolism in cognitively normal and dementia subjects. In conclusion, both the I2I network and the linear model could offer valuable prognostic insights, guiding early intervention strategies to preemptively address anticipated declines in brain metabolism and potentially to monitor treatment effects.