I know that Moya gave birth to Talyn and that this was the result of Peacekeepers trying to make a Leviathan gunship, presumably by some sort of artificial means. But how would Leviathans normally reproduce?
I think we can only judge from "They've Got a Secret" and "The Hidden Memory" and from what little evidence is added by "The Way We Weren't".
In the former, D'Argo falls down a shaft and accidentally breaks a Peacekeeper shield, releasing scads tiny biomechanoid "cells" (for lack of a better word) that apparently get Moya pregnant. These cells were confirmed by Zhaan to come from Moya herself. The baby grows inside a special incubation area until it is mature enough to "be born" and free themselves from the parent. The difficulty with Talyn was that he was not a normal Leviathan.
We later find out that the shield was placed there to sabotage the gunship project.
The upshot of all this is that Leviathans appear to reproduce asexually, since the biomechanoid "sperm" that impregnated Moya came from her as well.