Yes, I had this issue too. There is a bench with a wooden plate thing on the otherside of the farm plot. You just have to place a kobold who is pregnant on that machine and they will lay an egg. They must also have energy to produce eggs.
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To be honest, one kobold cumming into the player character should be enough to become pregnant. Then you just have to go to the breeding station and interact with it like you would with every other interactable. Do note, that you must have energy to interact with it, so make sure the energy bar is filled