While there are some translations that exist like "अलेक्जेंड्राइट" (pronounced "Alekzandrait"), these are not commonly used and aren't considered official names.
The gemstone is usually referred to by its English name "Alexandrite" or sometimes as "अलेक्जेंडर का पत्थर" (pronounced "Alekzandar ka patthar"), which translates to "Alexander's stone" but this is also not very common.
It's important to note that the Hindi language doesn't have a strong tradition of naming gemstones like some other languages.