Buconjić, who was born in Drinovci, Herzegovina, during the Ottoman rule, joined the Franciscans in 1851 and after a year of novitiate, he became a full member of the newly established Franciscan Custody of Herzegovina in 1852. He attended the University of Ferrara at the in Austrian Empire, and was ordained a priest there in 1856. He then lectured at Antonianum in Rome between 1860 and 1866, when he returned to Herzegovina, where he lectured at the seminary of the Franciscan friary in Široki Brijeg. Buconjić became a chaplain in 1871 and in 1873 a parish priest in Drinovci. He was elected Custos of the Franciscan Custody of Herzegovina 1874, a position he held until 1879. During the Herzegovina Uprising of 1875–1877, Buconjić with Bishop Anđeo Kraljević helped to pacify the Catholic population and supported the annexation of Bosnia and Herzegovina by Austria-Hungary. In 1875, Buconjić published a booklet titled "The Main Reasons for the Uprising of the Christian Population in Herzegovina" and sent it to the representatives of several European countries, which raised awareness among the European powers regarding the position of Christians. At the Congress of Berlin in 1878, it was decided Bosnia and Herzegovina would be occupied by Austria-Hungary. Buconjić was an important figure in Croatian politics during the Austro-Hungarian rule in Bosnia and Herzegovina.