One of the challenges in a viral pandemic is the emergence of novel variants with different phenotypical characteristics. An ability to forecast future viral individuals at the sequence level enables advance preparation by characterizing the sequences and closing vulnerabilities in current preventative and therapeutic methods. In this article, we explore, in the context of a viral pandemic, the problem of generating complete instances of undiscovered viral protein sequences, which have a high likelihood of being discovered in the future using protein language models. Current approaches to training these models fit model parameters to a known sequence set, which does not suit pandemic forecasting as future sequences differ from known sequences in some respects. To address this, we develop a novel method, called PandoGen, to train protein language models towards the pandemic protein forecasting task. PandoGen combines techniques such as synthetic data generation, conditional sequence generation, and reward-based learning, enabling the model to forecast future sequences, with a high propensity to spread. Applying our method to modeling the SARS-CoV-2 Spike protein sequence, we find empirically that our model forecasts twice as many novel sequences with five times the case counts compared to a model that is 30× larger. Our method forecasts unseen lineages months in advance, whereas models 4× and 30× larger forecast almost no new lineages. When trained on data available up to a month before the onset of important Variants of Concern, our method consistently forecasts sequences belonging to those variants within tight sequence budgets.