In his book East Belfast (Images of Ireland) Keith Haines writes that the Belmont estate gave the district its name. Belmont House was built in the 1830. It was originally the home of solicitor Alexander Montgomery, and later of Sir Thomas McClure MP, who lived there c. 1858-1890. He had a considerable influence on the development of East Belfast, particular along the Belmont Road (Haines, Knock, Knock: who was there? pp. 38-44). He sold the 70-acre estate in 1890 to the trustees of Henry James Campbell in order to found a school - namely Campbell College.
The name comes from the Old French bel mont, "beautiful mount", which is a common place-name in France (the modern French would be beau mont, also a frequently occurring name). There was a trend for house-names from French in the 18th and 19th centuries, such as Ormeau, Sans Souci, Montpellier and Belvoir.
The church referred to appears to be Belmont Presbyterian Church, situated at the northern end, actually on Sydenham Avenue. It was founded by Sir Thomas McClure, who gave 2 acres of land at a peppercorn rent in 1860 for the building of a church, manse, school and porter's lodge. The church opened for worship in 1862. McClure's sister Eliza presented a bell at the first service (Haines, Knock, Knock: who was there? p.42). The building at the corner of Belmont Road and Belmont Church Road is not a church, despite appearances. It is Belmont Tower, formerly Ferguson Memorial School.
“The surveyor having reported that there were two streets in Knock added area known as Church road Resolved that the street running from Belmont Church to Upper Newtownards Road be named Belmont Church Road and the street running from Sandown Road ‘Knock Church Road’". Minutes, Improvement Committee, 24th January 1900.“