A quarrel between friends, when made up, adds a new tie to friendship.Attr. St. Francis de Sales
Google Books got me close. An 1829 book, The Beauties of St. Francis de Sales, has the potential quotation pretty much as given, but in indirect form: “He rather preferred a contrary maxim; and said, that a quarrel between friends, when made up, added a new tie to friendship: as experience shews, that the callosity formed round a broken bone, makes it stronger than before.”
The part about the broken bone gave me enough additional info to track down the source, which is The Spirit of St. Frances de Sales, written by the saint’s friend Jean Pierre Camus. The quotation is given there in indirect form as well: “He did not admit the maxim of the world: ‘We must not trust a reconciled enemy.’ In his opinion the exact contrary of this dictum is more in accordance with truth. He used to say that ‘fallings out’ in the case of friends only serve to draw the bonds of friendship closer, just as the smith makes use of water to increase the heat of his fire. He added, as a well-known fact in surgery, that the callosity which forms over a fractured bone is so dense that the limb will never break again at that particular place. Indeed, when a reconciliation has taken place between two persons hitherto at variance, it is almost certain that each will set to work, perhaps even unconsciously, to make the newly-cemented friendship firmer. The offender by avoiding further offence, and atoning as far as possible for what is past, and the offended person by endeavouring in a truly generous spirit to bury that past in oblivion.”
So, given slight alterations along the way, the quotation is authentic.