Veteran actor Sam Elliott (and his iconic brush mustache) joins the surreal world of “Family Guy” this Sunday as recurring character Wild Wild West, a horseback-riding cowboy cousin of deceased Mayor Adam West, played by the real-life “Batman” star who died in 2017.

In his introductory episode (9:30 p.m. on Fox), Wild is recruited by Peter (Seth MacFarlane) to run for mayor of Quahog against local librarian Elle Hitler.

“My daughter has been a fan of [‘Family Guy’] for a number of years, and it’s because of her that I’ve seen a lot of episodes,” says Elliott, 76. “To be honest, I don’t wait up every week to watch, but I’ve certainly been aware [of the series]. You’ve got to be living in a hole somewhere … There’s no escaping ‘Family Guy.’

“There are a lot of super-intelligent, nice people that produce this and a few friends of mine looked at me and said, ‘Really? You’re gonna do that?'” Elliott says. “They just don’t get [the show]. My contribution is pretty much playing myself … It’s not like they wanted me to come up with a straight ‘character’ voice, so to speak. I’m leaning on the southwestern thing a little bit. Beyond that, I’m playing it fairly straight and letting the jokes take care of themselves.

“There’s an old saying, ‘If it ain’t on the page, it ain’t on the stage,’ and I’ve always been a believer in that.”

Elliott is no stranger to voiceover work; his honeyed, Western-tinged tones grace a variety of commercials (including Coors beer and Ram Trucks) and he snared a 2013 Emmy nomination for playing a (claymation) version of himself on “Robot Chicken,” the long-running Adult Swim animated series co-created by Seth Green.

“That’s what really started this,” he says. “It started a lot of things in my life and for my career. I ended up getting a couple of other films [because of ‘Robot Chicken’] and I did a film with Lily Tomlin [‘Grandma’ in 2015] because of it.”

He says that, when he was approached about joining “Family Guy,” producers wanted him to play himself. “I wasn’t interested in doing that,” he says. “Adam [West] had done that and he did it in spades, and that was like a bridge too far for me. I couldn’t imagine being [myself] week after week.

“So they kicked it around and came back with this brilliant idea for Wild Wild West that I couldn’t say no to,” he says. “It’s interesting filling that void, shoes I would never even think about filling … I had to be somebody else — and I’m grateful I happen to be a relative of Adam’s now.”

Twentieth Century Fox Film Corporation

Twentieth Century Fox Film Corporation

Twentieth Century Fox Film Corporation

Twentieth Century Fox Film Corporation

And, while “Family Guy” often ventures into risqué territory, Elliott says there was one scene where he balked a bit at a few lines of dialogue.

“I do have a certain amount of integrity, and if it’s distasteful, no thanks,” he says. “There’s a sequence where [Wild] talks about his background, what qualifies him for running for office, and he’s talking about raising his [16] daughters … and there were a couple of lines that could’ve been taken absolutely the wrong way and I had no interest in doing that.

“When talking to the producers they saw where I was on that and said, ‘No problem’ and they changed it — and the change ended up being funnier.”

Elliott also gets a chance to sing in Sunday night’s episode when Wild sits around a campfire and strums an acoustic guitar.

“That’s really me. I’ve sung my entire life and my mom dragged me to sing in a cherub choir when I was 4 years old in Sacramento at the Congregational Church, and that began a lifetime of singing,” he says — adding that he went to high school with the guys who formed the Kingsmen, who had a No. 1 hit in 1963 with “Louie Louie.”

“I used to sing with those guys at dances,” he says. “I wasn’t a member of the band but they would back me up because we all went to the same school.

“There was a point in time where I thought my career would be in music and not in the acting game. I never went after it,” he says. “I think I’m a little afraid of it.”