Is this something new, or did the headphone work fine for a month, and then suddenly quit working? If that is the case, I would return the guitar and expect an exchange. After only one month this should be completely within warranty.
With my Yamaha Silent guitar I had an issue where some earbuds don't work. It was a while ago and I usually don't play the silent guitar with headphones. The issue with one pair of ear-buds has something to do with the length of the sections of the headphone prong. By very slightly pulling the headphone jack in or out of the socket I could get that set of earbuds to work.
If I recall correctly (which I may not be), it was my Apple Iphone earbuds that caused this problem. The pair of cheap earbuds that the airline gives you worked fine, and my Bose headphone worked fine.
Final question - which may be dumb - do your headphones use batteries that require changing?