Multiple contact forms - Different recipients
But, is it possible to do this when there is another contact form on the site? 


Yes?


  If so, what do I need to add to the department dropdown code?


The dropdown code shouldn't have anything to do with it, if I'm following you correctly.

Both forms just need to have a toEmail hidden input as specified in the docs: https://github.com/pixelandtonic/ContactForm#dynamically-adding-email-recipients-requires-craft-25