The story in question goes as follows: a private investigator is sent to find a missing girl who had a thing for him and was trying to break into Hollywood. He finds a painting of a golden city in her apartment and takes it home. He gets sucked in and is transported to a marvelous golden city where stuff is weird. It's heavily ancient South American inspired with new tech. He finds singer lady there and they go home after adventure. I swear the world/country was called "Medula" or some such similar.
The Portal in the Picture aka Beyond Earth's Gates, a novel by Henry Kuttner and C. L. Moore, first published in Startling Stories, September 1949, available at the Internet Archive. You might have read it in the 1987 paperback collection The Startling Worlds of Henry Kuttner.
The otherworld city is called Malesco. It's described in the story as rose-red rather than golden. I don't know if there is anything South American about it. The Malescan language is derived from Latin, so there's that.
The hero and first-person narrator, Eddie Burton, is a Broadway actor, not a private investigator. The girl, Lorna Maxwell, is a nightclub singer trying to break into Hollywood.
Eddie did not get the picture from Lorna. He inherited it from his Uncle Jim, who used to tell him bedtime stories about Malesco and even taught him the language:
The air flickered again. I squinted and shook my head. This was getting a little alarming. It couldn't be the Scotch. It never happened outside the apartment. It never happened unless I was looking at that particular wall.
There was a Rousseau picture on it, Sleeping Gypsy, something Uncle Jim had left me along with the apartment. I made a great effort to focus on the blue-green sky, the lion's blowing mane, the striped robe of the black man on the sand.
But all I got was a blur. And then I knew I must be drunk because a sound seemed to go with the blur, a roaring that might have been the lion except that the lion had entirely vanished and I seemed to be seeing a dome of shining rosy-red light that moved like water.
Lorna visits Eddie in his apartment and gets sucked ino the portal. Eddie is suspected of murder but of course they can't find the body. Then Eddie gets sucked into the portal too and finds himself in Malesco, where Lorna has been enhanced and is a big star. After some adventures he returns to New York with Lorna.