Not always. My mother started laboring in the AM after she woke up (?~11am) and I was born in the afternoon.
But there is a study that looked at this:
https://journals.plos.org/plosone/article?id=10.1371/journal.pone.0198183 and found that for spontaneous births, nighttime labor/births are more common.
Just over half of all births in our data set are spontaneous births following spontaneous onset of labour. These have a roughly sinusoidal pattern, and are most likely to occur between 1:00 and 7:00, with a peak around 4:00, and a trough in the afternoon.