The actors took to Instagram to reveal their beloved dog had died on Friday
On Monday, Moyer, 54, took to Instagram to confirm their "beautiful" pet had "passed over to the other side" on Friday.
"Wherever that is, whatever that is, it was mercifully gentle and he was surrounded by all of his loved people & of course, Dave," the actor wrote, referencing their other rescue dog.
Moyer continued, "Over the past few weeks he has been slowly leaving us; we would walk a few steps and then he would lay down."
"Like in this pic from a walk last week," he continued. "I would lift him, he would walk a few steps, then slowly drop to the floor & try to sleep again."
Moyer added that it's "hard to describe the odd ball that Banjo was."
"He asked for nothing. From the outset. He always had his back turned to the wild world. He would seek out the nooks and crannies of a room and hide under tables and desks."
Moyer and Paquin, 41 — who tied the knot in 2010 — rescued Banjo "from a Puppy Mill in 2009," the Devil's Knot star explained.
"I always wondered what had happened to him back then. When I saw him that first time at the adoption drive, all the other dogs were yapping and jumping up demanding to be adopted, but Banj was in the corner in a little heap with his body turned away and asleep. Tiny little rag doll," he continued.
Moyer recalled how when he went over to look Banjo in the eye at the time and "he barely looked up at me."
"We were told he was part Louisiana Catahoula leopard dog. (Look it up - its’a thing) he was like a Dingo from the front with a greyhounds back legs and a leopards spots," he added.
"One boss eye which meant you never quite knew if he was quite looking at you. There would be differences of opinion about the origin of his name, me romanticizing that we called him Banjo because he kinda looked like the genius Banjo player in Deliverance but the true story being that Anna loved the song ‘Old Susanna’ which has the lyric ‘Oh, I come from Alabama / With my banjo on my knee / I’m going to Louisiana / My true love for to see.' It felt perfect somehow that he was our true love and he had found us. Our Louisiana lad."
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"He was such a good good boy. So loved. So odd. And will be so so missed," Moyer — who shared Banjo and Dave updates on their @daveandbanj Instagram account — wrote.
Paquin reposted the message, with the caption: "💔 I really can’t word it any better than Stevo did. 💔"
Moyer and Paquin met on the set of HBO's vampire drama True Blood in 2009. They married in 2010 and welcomed twins — daughter Poppy and son Charlie — in 2012.
In January, Moyer told PEOPLE that despite working with his wife multiple times over the years, he still loves it.
"It's just the best," Moyer said.
"I mean people ask us a lot, 'Isn't it weird working with your spouse?' And I'm like, well, it's how we met," he continued. "We were on set every day together for seven years. And so actually when we're not working together, it feels like I miss her. If feels weird."
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