The daughter of Voldemort and Bellatrix is revealed in The Cursed Child.
In a scene in the play, she claims that she was born in Malfoy Manor before the Battle of Hogwarts (note that, in the interests of protecting spoilers, I've redacted the character's name):
[Daughter]: I am from the future. The child of Bellatrix Lestrange and [Voldemort]. I was born in Malfoy Manor before the Battle of Hogwarts.
Harry Potter and the Cursed Child Act Four, Scene 11
Although the daughter of Voldemort is necessarily an unreliable source of information, in this instance (because of time travel shenanigans), she is talking to her father and has no reason to lie.
If any kind of surrogacy was used, it is not discussed.