Troi gives birth in the episode TNG: The Child. The lights are dimmed and she's sitting in a sort of 'birthing chair' with stirrups. Since they've had at least a few hours to prepare for her arrival this is, we can reasonably assume, the normal birthing procedure for human women in the Federation.
The original screenplay also indicates that pain, pain relief and trauma are all still part of the normal process of giving birth in the 24th Century along with ensuring that the mother remain awake and alert in order to experience the birth.
PULASKI: (continuing) I'll give you something for the pain. It won't
in any way diminish the experience.
[They now have Troi in position in the birthing chair. Worf and the
security team enter and take up their positions as Pulaski and her
Assistant prepare for the delivery.]
[WE WILL LAY THIS OUT, BASED ON HOW A CHILD IS BORN ON THE
TWENTY-FOURTH CENTURY ENTERPRISE.]
PULASKI: I have delivered dozens of babies. But none like this. There was no pain, no trauma. It was effortless for both of them.
We see another Federation birth (albeit from an earlier era and on-planet) in Star Trek V. Spock's mother gives birth lying on a big rock in a cave with a priestess to act as midwife. The film script indicates that there's no modern technology involved and no pain relief.
[They turn and find themselves beneath the hanging stalagtites of a cave. Torchlight throws weird shadows on the walls. The shadows depict a woman in labor, attended by a Vulcan High Priestess. Ceremonial drums pound a throbbing pulse. The woman lets out another scream. It reverberates throughout the cavern.]
[Spock's mother, the young earthwoman AMANDA, lies on a rough pallet, her legs spread beneath her robes, her distended stomach lifting. Her only attendant is THE HIGH PRIESTESS who intones the Vulcan birthrights.]
You might wish to note that while Fetal Transport does exist, it's (presumably) generally avoided because it can lead to post-birth complications in the child. In the single case that we see it on-screen, the transport apparently causes young Naomi Wildman to suffer a deadly (albeit treatable) condition.
WILDMAN: Is she all right?
EMH: The transport caused a slight hemocythemic imbalance, but we'll stabilise her cell membranes with osmotic pressure therapy.