Is it accurate to say to God that he ‘has no birthday’?

There is a song I like from a source that I trust, called ‘God of Wow ’ that has as its very first line ‘You have no birthday’ and that stops me from sharing or using it. My objection is that although it is true that God is eternal and birthday-less, but it seems to me that the external God did take on having a birthday because of the incarnation where Jesus was conceived and birthed. I understand what the song means, but is this a legitimate phrase? I am asking from a Nicene-Christian perspective.