How do you quote someone whose native language is not English? Let's assume that the person make several grammatical mistakes, how should you quote them, do you quote them verbatim or do you quote them after fixing the mistakes, or do you put the mistake and put the words that should have been used instead in parenthesis? What's the gold standard here?
Here's an example:
"The country's relation to China has been warming up in recent times." said John Doe.
In the case where someone is speaking poor English, it's generally advised to quote exactly as spoken with the addition "[sic]" which is Latin for "Thus" or "just as" and is generally a shorthand for "sic erat scriptum" (thus was it written). It's generally meant to denote that the author is transcribing the quoted dialog and that the grammatical error was made by the speaker of the quote and not the writer.
Fixing a grammatical error made by someone you were quoting can result in a changed meaning or misattributed something the speaker did not say.
It is not actually up to you how you cite a source, but it is prescribed by the style guide relevant to your field (e.g. MLA, APA, Chicago, etc.). In the style guides I'm familiar with (MLA, APA) it is a requirement that you cite your sources verbatim. The only change you must make is to append "(sic)" or "[sic]" (depending on style guide) to passages that contain obvious linguistic errors, to make it clear that they aren't errors on your part, and in some cases a translation, if the cited text is in a foreign language. Otherwise, you mustn't change the text that you cite! You can paraphrase the source in your own words, though, and of course avoid grammatical mistakes in your paraphrase.