Are there any connections to religion to the Birth of Venus by Sandro Botticelli?

No, there is nothing religious about the Birth of Venus. It depicts the goddess Venus coming out of the water at the moment of her birth. The god Zephir and the nymph Chloris blow her to shore. On land, Primavera awaits with a blanket to enfold her.

I would say that it had references to pagan religions, particularly that of the Romans since Venus was a Roman goddess.<<<<<<That is false. Pagan religion was not being practiced at the time Botticelli was alive. Birth of Venus actually depicts mythology and religion at the same time. Botticelli needed to do this to please the people who studied the humanities, but also people of faith. While the image depicts Venus' birth from the sea(Venus was formed from the foam created from her father's genitals being thrown into the sea), but this can also coincide with the Christian practice of Baptism. Because Venus is a 'divine' figure emerging from water, the story can correlate with a divine figure's Baptism(leading to a connection to Jesus' Baptism).

The theory that religious beliefs and humanistic studies can coincide is called Neoplatonism. For more information on this topic, visit the link to website below.