In fact the doctrine of the Trinity says that God has some level of distinction within himself. God is personal. There is enough distinction within him to allow him to experience love. This means there is within God both lover and beloved. He experiences both the creative love of the Father, and dependent love of the obedient Son.
Personal relationship implies communication.
I don't exactly think the Trinity uses prayer as its internal communication. Prayer is uniquely the way humans communicate with God. But I do think there is something like communication within the Trinity. When we say that Jesus shows us God, there's obviously some sort of indirectness. He is the visible image of the invisible God.