One possibility is to look at the minutes of the kirk session for the parish to see if there are details of the father there. If they survive they would be available to read on-line in the National Records of Scotland in Edinburgh and in some archives elsewhere in the country.
It might be useful, however, to look at the 1871 census to see if there are any suspect John Falconers in the area. The mother, of course, might have conceived the child while working, perhaps as a domestic servant, away from home, but returned there for the birth.