You may need a more specific location within Luxembourg but otherwise, you have a good chance of finding a birth certificate with the birth date and birth name. But that's assuming a birth certificate was even issued and assuming it's available to the public. You would have to look into the availability for this specific location.
Remember that official, government mandated birth certificates were not always issued in history - they are relatively modern, mostly introduced in the late 19th/early 20th century depending on the location. However, recordings of births or baptisms (not the same as a birth certificate) by churches or local governments have been around for many centuries. For example, Family Search has a birth/baptism index collection for Luxembourg 1662 - 1840: https://www.familysearch.org/search/collection/1520578
- these are not birth certificates, just birth/baptism recordings. If your date is more recent than this, from the 20th century, there may have been a birth certificate.
You may also find these Luxembourg specific genealogy sites helpful: http://www.familux.org/genres.htm