Can I specify a poisson prior somehow?

No and maybe. A Poisson prior requires that the parameter over which you are placing the prior be discrete. Stan does not support discrete parameters. It turns out there are very good reasons for this, and even modeling languages that do support discrete parameters often can be made to sample more efficiently by reparameterizing the model to avoid discrete parameters. This process is known as marginalization. And it is likely that we can write down a marginalized version of your desired model that encodes the same prior and everything. If you can write out your desired model in mathematical notation, we can probably help you fit it.

To add on to @jscolar’s excellent answer, there is a Stan User’s Guide chapter which also gets into the what, how, and why of marginalization: