The journey of a thousand miles begins with a single step.

Yes, with one translation note. Laozi does say this, in Tao Te Ching chapter 64: 千里之行，始于足下. Literally, a journey of a thousand li starts beneath one's feet. James Legge in 1891 translated the clause as "the journey of a thousand li commenced with a single step." The sentence everyone repeats — a thousand miles, beginning with a single step — is that line, with "li" changed to "miles" and the verb brought into the present. The picture is the same one. The unit and the tense are not.

A li is not a mile, and Legge's verb is "commenced", not "begins". The verdict is still verified, because the Chinese says what the proverb is getting at: the long journey starts from the ground already under your feet. Cite chapter 64. Prefer "a thousand li" when you are quoting Legge or the text itself. "Miles" is the later convenience. It does not make the line a fake, and it does not make the modern sentence word for word Legge. Say which English you are using, and keep the chapter number beside it.