Pregnant Woman Side Quest - Mega Man Legends
The Pregnant Woman Side Quest only becomes available once you've unlocked the final dungeon in the game, the Main Gate. In order to start this quest you'll need to visit the Apple Market in town and speak with the man who is running the Vegetable Store (pictured above). He will make a comment about his wife who went out for a walk and has yet to return - this is the flag you need to find his wife.
You can find this man's wife in the southern portion of the Cardon Forest, the exact location is an alcove in the western portion of this area. I'd recommend calling up Roll and having her take you to the Cardon Forest - this is the same location that she parks the Flutter airship. From here you'll want to go south until you reach the Ruins #2 dungeon.
The alcove that I am talking about is directly across from the Ruins #2 dungeon that is in this area. A screenshot of where you can find the pregnant woman is below, she's leaning against one of the walls in the area catching her breath. When you approach her you'll be given the option of taking her to the hospital or waiting to see if she gets better - take her to the hospital to update this quest.
Back at the hospital you'll get a short scene after which you'll want to leave the hospital and enter it again to update the quest. Now all that's left is to talk to the man in the waiting room to get your reward. Your reward for completing this quest is the Sun-light item. Roll can turn this item into the Omni-Unit Omega Buster Part which is one of the best Buster Parts in the game.