Terreni was born in Livorno in 1739 and later studied in Florence with the English 'dilettanti' Ignazio Hugford (1703-1778). When part of the Uffizi was damage by fire on 12 August 1792, the Grand Duke Leopoldo de'Medici commissioned Terreni, along with Giuseppe del Moro and Giuliano Traballesi, to paint a new ceiling for the Gallery. The success of these frescoes was so great that Terreni was soon afterwards asked by the Grand Duke to redecorate the Sala del Buon Umore in the Villa di Poggio Imperiale. Later, he worked on the Sala delle Nicchie in the Palazzo Pitti, Florence. Between 1771 and 1773 he painted the Cupola of the Santuario di Montenero in his native city. A view of the Duomo of Livorno, dated 1774, is in the church of San Ferdinando in Crocetta. In 1779 he painted the cupola of the church of San Leopoldo in Livorno. The frescoes in the cupola of the chapel of the Sacrament at the Duomo of Livorno, finished in 1798, and destroyed during the Second World War, were considered to be his best work, C. Venturi, Pittori livornesi: Giuseppe Maria Terrini, in Liburini Civitas, VI, 1933, p. 39, illustrated. Terreni also worked in Pisa on the decoration of the Certosa di Calci.
Terreni is also known as a vedutista: in 1781 he dedicated to the Grand Duke Leopoldo de'Medici a volume of eighteen prints entitled 'delle più belle vedute dellà città e porto di Livorno', Venturi, op. cit., pp. 48-51, illustrated. He painted in 1783 a view of Livorno for the Grand Duke and another of Florence, now in the Galleria Palatina in Florence, (L. Salerno, I pittori di vedute in Italia (1580-1830), Rome, 1991, no. 82.1, illustrated). Two years later he signed a pair of views of Pisa now in the Palazzo Pitti in Florence. He produced with Francesco Fontani a series of aquatints for the Viaggio pittorico della Toscana in 1801-3.