I recently retired, I was told I could collect unemployment for one month, is this true?

Not true for 2 reasons. You have to be let go by the previous employer without it being misconduct. Retirement does not count. Also you have to be looking for a new job.

I work in a different department, but I've never heard of this before. I seriously doubt it's true.

Just apply

Always try filing. The worst thing that can happen is they say no. In the states eyes a retirement is same as quit.

You can apply but probably won't be eligible.