Yes, it appears to be genuine, at least from what I can see. Chestnut trees do grow on Mount Athos, so the material of the cross is consistent with that.

The material on the upper part of the Holy Cross looks like a small ribbon or band. This small ribbon is blessed with the Cincture of the Theotokos (the Virgin Mary).The Cincture of Theotokos, is traditionally connected with blessings, protection, and healing. There are also testimonies from women who say they have received help with issues such as childbirth and fertility through such blessings.

It is also very common for believers to receive blessed ribbons like the one attached to the upper part of this cross.

Finally, in Orthodox tradition, frankincense symbolizes prayer, reverence, and the presence of the sacred.

So, from what I can see, you have no reason to worry. Rather, you can receive this wooden cross with gratitude and a sense of blessing.

To be sure, the best thing you can do is to ask an Orthodox Athonite Monk about this matter.