Is there a way to ask another Civ to convert my cities, e.g. if I don't have a religion.
No.
You can send caravans to cities with the desired religion. The more caravans from one of your cities to several cities with the desired religion, the faster the city will adopt the religion (hover your cursor over the city bar and see the religious numbers and pressure).
You can declare war and capture missionaries if you see them running near you. The Great Prophet cannot be captured; he will be destroyed.