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“We have long ago paid the goblins of Moria,” said Thorin.

From chapter 1 of the book The Hobbit by Tolkien. What does it mean? Did the goblins pay for their murder or did the dwarves stop paying them thoughts?

Not a native speaker so some expressions I have to look up, this one I could not resolve. Just finished reading the first chapter.

The reference is to the murder of Thorin's grandfather Thror a few paragraphs earlier:

"Your grandfather Thror was killed, you remember, in the mines of Moria by Azog the Goblin."

So when Thorin says "We have long ago paid the goblins of Moria", he is speaking of revenge. The modern idiom (in American English, anyway) would be "We long ago paid back the goblins"; I don't know if the difference is due a British English idiom, a change over time, or both.

The revenge Thorin speaks of is the War of the Dwarves and the Orcs, more information about which you can find in Appendix A of The Lord of the Rings. There is at least one reference to it in The Hobbit itself, in Chapter 16 "A Thief in the Night", when Bilbo is meeting with Bard:

".... Dain, I may tell you, is now less than two days' march off, and has at least five hundred grim dwarves with him—a good many of them have had experience in the dreadful dwarf and goblin wars, of which you have no doubt heard."

(Of course, this reference at the time of writing was probably to an unspecified war or wars, whose details were later filled out more properly for The Lord of the Rings.)

He means that they metaphorically "paid back" (i.e. took revenge upon) the goblins for their role in Thror's death. At the Battle of Azanulbizar, where Thorin earned his name "Oakenshield" and his cousin Dain slew Azog the goblin king, the dwarves exacted their vengeance for the murder of the king of the Longbeards.

Per Tolkien Gateway:

The War of the Dwarves and Orcs began when Azog the Orc-chieftain of Moria captured and mutilated Thrór, King of Durin's Folk. Azog branded his own name in runes onto Thrór's severed head, then let Thrór's companion Nár escape so that all Dwarves would know that an Orc now ruled Moria. Full of righteous fury, Thrór's son Thráin II summoned a great army of Dwarves, including those not of Durin's Folk (Firebeards and Broadbeams from the Blue Mountains, and others from the far East of Middle-earth). For six years they systematically sacked the Orc strongholds of the Misty Mountains, until only Moria was left. There the Orcs that had survived the destruction had gathered to Azog.

...

The Dwarves were victorious, but half of their forces were dead or mortally wounded. The Orcs suffered even worse casualties, with ten thousand dead. After the battle, Thráin wanted to enter and reclaim Moria, the ancestral home of Durin's folk. However, due to their losses, the other Houses not willing to participate, and since Dáin had seen Durin's Bane beyond the East-gate, Thráin refrained from entering.

...

Although the Dwarves suffered heavy casualties, the battle would have lasting effects for the Orcs of the Misty Mountains. Their numbers were severely reduced after the battle and never fully recovered.

Thorin follows up his comment about having paid the goblins for killing his grandfather with a suggestion that they should wage another war agains the Necromancer, who was responsible for his father's death. However, Gandalf tells him that even all the dwarves together would not be powerful enough to defeat the Necromancer, who Gandalf (unlike Thorin) knowns to be none other than Sauron—the strongest surviving menace from the First Age.

While the common idiom for revenge is "pay back", I think that Thorin refrains from using "pay back" rather than the slightly strange "pay" in order to avoid their actions being seen as on an equal level with that of the goblins. "Pay back" usually refers to a kind of tit-for-tat, an eye for an eye action rather than annihilation (which was the target of the wars even though the dwarves did not achieve it). Thorin does not want to acknowledge the damage the goblins exacted on the dwarves first as being comparable to what the dwarves chose to do in reaction.