The brown spot is most likely a nevus (mole), and as far as I know there is no non-surgical method to remove such a nevus on the conjunctiva of the eye.
For your information, the surgery is relatively minor anyway, and is conducted as day surgery under local anaesthesia. It involves removing the spot and patching the area with a free graft of adjacent conjunctiva to minimise scarring.
Most of the time, surgery is only cosmetic in nature and so is not absolutely necessary. However, if there is documented growth of the lesion, or it is large and shows certain suspicious signs, then an excision biopsy for further testing could be necessary. If there is any concern about this, do consult with an eye doctor for further assessment.
