Why did Quint have the tattoo removed?

Firstly, I think we can assume that Quint had the tattoo before the USS Indianapolis sank so it was a 'celebration' of the ship, when she was afloat rather than a 'memorium' to the ship after She sank.

Quint, as you can tell from his reticence to tell the story, his subsequent obssession with sharks and the 'tone' he uses to tell the story (especially how he says 'bomb'), that the whole episode affected him greatly and didn't leave a positive impression.

I'd therefore suggest that he was embarrassed or traumatised to be associated with the ship and all She stood for, especially the delivery of the bomb.

For this, reason, he had the tattoo removed.

Some other resources:

The only non-existent tattoo on this list – as Hooper and Quint drunkenly compare scars on the Orca in Jaws, Hooper notices a scar on Quint’s arm. It is identified as a removed tattoo for the USS Indianapolis, a real life Navy ship sunk during WWII that was known for the amount of shark attacks inflicted upon the survivors.

It’s this story – Quint’s time stranded at sea told in an amazing monologue – that reveals so much about the character. We suddenly understand exactly where this man is coming from, that he experienced something that will never go away, no matter how hard he tries. His only hope is to face it head on, which explains his profession.

Really it’s the fact that he removed the tattoo, the fact that he didn’t want people to ask him all the time about it and not be reminded by it that shows a sliver of vulnerability in an otherwise hard-as-nails man.

From : Character defining movie tattoos

We learn this from the scene wherein Quint and Hooper are comparing physical wounds. It's all fun and games - a friendly competition - until Brody asks about one scar on Quint's arm. Turns out, it was from a tattoo removal. That tattoo in question was from his time in the Navy; more specifically, it was related to the Indianapolis. Of all the scars, this was the one that wasn't a laughing matter. Sure, the story was inherently grave and serious, but it wasn't ultimately about the Indianapolis. The story highlights that Quint left the experience (understandably) scarred mentally.

He tried to remove the memories when he had the tattoo removed, and perhaps his career choice was also an effort to exercise his demons. His profession? Shark hunter. Having survived but witnessed a huge number of shark attacks, he spent his life killing sharks himself. Rather than staying as far away from the ocean as possible, he went on some sort of revenge motif. Or maybe he hunts sharks because he knows it can be dangerous and he might finally suffer the fate he feels he should have suffered during the war. He seems to know right away how dangerous the shark in Jaws actually is, as evident by his asking price.

From : Jaws - all about Quint

The scene looks at the continual themes of loss, paternal mismatch and distrust of closeness as well as all of the above analysis. Quint believes that Hooper is the typical 'wet behind the ears' youth that he distrusts and the initial one-up-manship is designed to embarrass him and show his lack of understanding. Quint however then gains some respect for Hooper and concludes the scene by unpacking that the macho physical scars are actually only skin deep and the scars of 'lives lived' are the ones that hurt. Much like Ahab he realizes that the only way to move on is to meet his daemons head on rather than trying to 'cut out' a pain that will not go away. His refusal to accept help in this quest when destroying the radio is the final opening of the wound left by the sinking and begins the path to his eventual anagnorisis and release at the hands of the shark. The initial ordeal left him half a man, the climax does the same.