Coral Island

First of all you must become married to an NPC.
In order to marry an npc, you have to get their hearts to maximum, give them a dating locket (from blacksmith), get their hearts to maximum again, and then complete every "heart event" scripted scene with them to progress your relationship. Check the Coral Island wiki for whichever (eligible) character you want to shack up with. You also have to upgrade your house all the way.

Once Married
Your sweety moves in with you and you can give them a gift a day everyday. Keep giving them Pink Diamonds (you should be replicating them with slime machine) until you get the new maximum of 15 hearts. I only have 1 set of data on this marital transaction, but I couldn't find anybody else who had noted down exactly when things happened, so your mileage may vary but the steps work regardless of timing.

Summer 10 I was married
Fall 1 I maxed out my wife's hearts to 15
Fall 4 I got a prompt to have a kid
Fall 18 Kid 1 arrived
Summer 3 Have a kid prompt appears again
Summer 17 kid 2 arrives.

Achievement Unlocks when you go to sleep the day your 2nd kid arrives.

This and 1 other achievement was the only thing I had left that was actually unlockable so I would just pick up pink diamonds and go to sleep, vendoring any gifts my wife gave me. I talked to my wife at first before kid 1 because there were a couple new monologue options but it doesn't seem necessary. Just wait her out and she'll eventually spawn kids. Gestation for your goblins seems to be a bout two weeks. You can't interact with your kids and they just sleep all day. Once again you're here on Coral Island for a long time not a good time. And now you've raised a family.