Do transgender men have periods?

Only if they aren't on testosterone and still have a uterus.

Yes, some transgender men do have periods - if they have female reproductive organs and if their hormonal treatment hasn't stopped their menstrual cycles. Most trans men I know use menstrual cups so they don't have to worry about it too much, or some menstrual product companies like Thinx are making products or advertising to include products to trans men too.