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 1830s. 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, Bellevue and Belvoir.
The 1870 Belfast and Province of Ulster Directory contains a reference to Belmont Road. It is clear from the 1877 directory that the street is still being built.
The McClure Baronetcy, Lord Belmont in Northern Ireland blog