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We have this idiom-like saying in Turkish. The idea is that there are certain things, topics, etc., if one talks about it, it strongly suggests that he has no idea what he is talking about, else he wouldn't be talking about it (or bragging about it) at all. Though not a one to one analogy, it is like a man in the bar telling the chicks that he is a spy; if he was a spy, he wouldn't be talking about it.

Is there a similar saying or idiom in English?

There's a famous quote from George R.R. Martin:

"any man who must say 'I am king' is no true king at all."

Source: https://www.goodreads.com/quotes/753661-and-any-man-who-must-say-i-am-king-is

But that's just a quote from a novel, not a bonafide english idiom.

That saying is common in many languages, including English, and often attributed to a Chinese philosopher, Lao Tzu, believed to live around 600 BC, who is believed by many to be the author of the book "Tao Te Ching", a fundamental text for both philosophical and religious Taoism.

“Those who know don’t talk. Those who talk don’t know. Close your
mouth, block off your senses, blunt your sharpness, untie your knots,
soften your glare, settle your dust. This is the primal identity. Be
like the Tao. It can’t be approached or withdrawn from, benefited or
harmed, honored or brought into disgrace. It gives itself up
continually. That is why it endures.”

Tao Te Ching

Martin Manser, The Facts on File Dictionary of Proverbs (2002) reports that the following anglicized (from Italian) proverb comes closest to the sense of the Turkish/Chinese proverb that the poster asks about:

who knows most, speaks least Wise or knowledgeable people say little: Don't be misled by her reserved manner—who knows most, speaks less. The proverb was first recorded in 1666, in an Italian proverb collection.

The earliest instance of this proverb that a Google Books search turns up is from Petroleum Times, volume 21 (1929) [snippet view]:

The menu card was very unique, being cast in the shape of a petrol pump. It was full of humour and quaint sayings. Underneath the toast list one read: “He who knows most speaks least,” consequently, the speeches were commendably brief.

Manser also has an entry for the exact phrase that the OP asks about (albeit in reverse order):

those who know don't speak; those who speak don't know Those who talk most volubly are usually those who know the least about the subject in question: The more you say, the more you show your ignorance—have you never heard the proverb "Those who know don't speak and those speak don't know"? Of ancient Chinese origin, the proverb was first recorded in English in 1948.

Actually, a sermon titled "Those who know, don't speak! those who speak, don't know" is announced for Christ Church in the [Moama, New South Wales] Riverine Herald on February 8, 1941. One can only hope that the sermon was short and to the point.

Whether either of these apothegms qualifies as a truly naturalized English proverb at this point is subject to individual interpretation. Neither is especially abundant in English usage, as far as I know, but both have appeared in English publications, going back at least seven decades.