Short answer: This is a bit of English folklore about country boys come to London, the tale of Dick Whittington (likely influenced by the Book of Revelations) but like folktales always do, it morphs to fit new specifics. So it's less about what immigrants thought than the way that Anglos viewed immigrants' aspirations through the prism of their own popular stories.
Discussion:
This appears to be more a bit of hyperbole used to describe immigrants' ambitions rather than something actually believed. The specific reference appears to be English in origin, its something immigrants may have heard, but not something they invented. The notion of the United States as a land of wealth, and the use of gold as a metaphor for that wealth can be found in cultures as far flung as Eastern European Jews, for whom America was the goldene medina ( = "golden country") and the Chinese Gumshan ( 金山 = "Gold Mountain").
In the case of the Chinese, the meaning wasn't to the US in general-- they were referring specifically to California, where they came during the Gold Rush, and often even more specifically San Francisco.
When you hear it said that the immigrants "thought the streets were paved with gold"-- perhaps someone believed that literally, but for the most part it appear to be an English language description, probably influenced by Revelations 21:21
"And the twelve gates were twelve pearls; every several gate was of one pearl: and the street of the city was pure gold"
This seems to have been a metaphor that was reasonably familiar in the English speaking world, before the Gold Rush, and indeed before widespread immigration to the United States.
The notion of "cities of gold" is much older than the 19th century-- so much gold came out of the Americas for so long that the reality of Potosi and the stories of El Dorado (much touted by Sir Walter Raleigh at the end of the 16th century) and the Seven Cities of Gold were very much a part of popular culture. Checking Google's Ngram data, the phrase "streets paved with gold" was known in the early 19th century, but really spikes in frequency in 1839-40, corresponding to the beginning of the California Gold Rush; it seems to have been a preferred trope in English language press, and this usage precedes immigrants' formulations.
The metaphor also applied to other places-- here's a description from the Australian Gold Rush, where the most literal case of "streets paved with gold" did allegedly occur. This 1853 description on of several cases where there's a claim of "gold in the streets" -- this is described by an Englishman, G.B Earp:
[G]old was now found a mile from Geelong- at Mount Disappointment - at the Pyrenees-a valuable copper mine was found at Deep Creek, whilst exploring for gold-and finally, the people of Melbourne began to break up the streets, which were macadamized with quartz pebbles brought from the gold localities. Gold, as might have been expected, was found, so that Melbourne may fairly lay claim to the honour of having had its streets paved with gold.
Similarly dubious, in 1839 an English traveller in Transylvania tells us
I dare say everybody has heard of streets paved with gold ;_ but I must confess I had always believed it a romance ; here, however, it was a serious reality. In fact, the road was formed of stones from the nearest rock, which we already know contains gold, and as it had been raining during the night, it was no wonder that the water should have washed away the lighter particles which had been crushed to dust under the feet of the passers, and left the heavier ore glittering in the sun behind.
And even earlier in 1828, an English writer J. C. Whitehead comments
The vulgar belief that the streets of London are paved with gold, still prevails in some parts of the country.
He is here referring to a bit of English folklore, the story of Dick Whittington and his cat; he is a country boy enticed to London by rumors that the streets were paved with gold. This very old story became popular in the 19th century, and is likely the source of all these references to "streets paved with gold". Whittington was a real person, a mayor of the City of London in the early 15th century, famously rich, a country boy who came to the big city and struck it rich-- while much of the folklore about him is invented, there was a real person to inspire it.
So we can say that the metaphor was already familiar in the English speaking world, and while you can find language in immigrant communities that approximates this claim, the roots of this trope are both older and more English than the immigrants to whom the idea is often attributed.
Sources:
Earp, G. B. (1853). The Gold Colonies of Australia, and Gold Seeker's Manual. United Kingdom: Routledge.
Young America and Australian Gold: Americans and the Gold Rush of the 1850s
Gold Mountain Guests: Chinese Migration to the United States, 1848-1882
The Enigma of the Isle of Gold
Paget, J. (1839). Hungary and Transylvania: With Remarks on Their Condition, Social, Political and Economical. United Kingdom: J. Murray.
Whitehead, J. C. (1828). King James the Second, or the Revolution of 1688: a dramatic poem [in five acts]. With ... notes. United Kingdom: (n.p.).
"Turn Again, Dick Whittington!": Dickens, Wordsworth, and the Boundaries of the City