I am in the midst of writing a play based on the Persian poem Layla and Majnun by Nizami Ganjavi. To do this I plan to make use of quotes from an English translated version of the poem. Archive.org has a copy of the poem which it lists as of being Public Domain Mark 1.0. From my understanding, this would indicate that using quotes from it will be allowed (with attribution) without contacting the original copyright holder. Is this correct?
If the translation is in fact in the public domain, and free of copyright, you may freely quote as much of it as you please. You are not even legally required to provide proper attribution, although it would be good practice to do so (and in my view morally wrong not to).
However, the link you provide says that the translation was originally published in 1966, if I have read it correctly. It is not clear who posted this copy, or how that person determined that it was out of copyright. 1966 is not long enough ago that such a work would be out of copyright by expiration, although there are various other ways that a work can lose copyright protection.
If whoever posted the online copy was incorrect, and the translation is still in copyright, you might well be infringing copyright if you quote it extensively, and could be sued. You wight want to contact the translators to verify the copyright status.