Abraham is located at Finch farm with his wife Abigail and his two sons, Daniel and Jake. There tends to be conflict between Abraham and Jake, which is most likely the reason his son ran away and joined the Forged.
He inherited an heirloom, a sword that has been in his family since pre-War times. Abraham asks the Sole Survivor to retrieve it in Out of the Fire.
↑Abraham's middle name is given both as Lowell and Francis: Abigail Finch dialogue: "That's why we made that rule in the first place. And I might remind you, Abraham Lowell Finch, that you agreed to it!" and Abigail Finch dialogue: "Abraham Francis Finch that is enough."