Shipbuilding first began in Belfast in 1791 when William Ritchie, with the help of his brother Hugh, built the first shipyard. A year later they launched their first ship, Hibernia, and over the next 20 years built more than 30 ships. From this start, shipbuilding in Belfast continued to grow throughout the nineteenth century. William Dargan built Dargan's Island (later renamed Queen's Island) in 1841 as a site of leisure activities though with the continued industrial growth of Belfast the island was quickly repurposed. Thompson and Kirwan yard was the first build on Queen's Island in 1851. Harland and Wolff followed suit by establishing in 1861, and became the major shipbuilder in Northern Ireland as well as much of Britain.
There is the brief narrative for you. To give a sense of the growth of both Belfast and it's industrial output I am going to quote from another answer I gave on this sub:
In 1800 the population of Belfast was roughly 20,000 but grew more than ten times to 230,000 by 1880. During this same period Dublin merely doubled in population. Primarily, this was due to the development of the linen industry, different from the wool industry of the South. There are a variety of reasons for the success of linen, none of which are significant to the question at hand (such as tenant rights, British suppression of the wool industry during the seventeenth century). A few other key developments included the short growth of the cotton industry between 1800-1820, which brought in a large number of Protestant immigrants, and the cotton shortage caused by the American Civil War. The cotton shortage pushed Ulster linen to the top of the market due to it's ready availability.
Belfast is built on the river Lagan, a waterway that was not initially capable of large shipbuilding. However, with the success of Ritchie and other builders that followed him, as well as the ever-expanding industrial production of Belfast, the Harbor Commission began renovating the Lagan to allow for shipbuilding to prosper. These renovations trudged up a great deal of the soot and mud and allowed Dargan to construct his Island. This construction not only gave shipbuilders the land/means to expand but proved a city-wide support for the burgeoning industry.
To be blunt, there really is no clear reason as to why Belfast became the shipbuilding hub that it did. From a purely industrial standpoint the success of Belfast's linen industry allowed the city to flourish, and necessitated diversification, one of the key forms of which was shipbuilding. But to quote the historian John Lynch, shipbuilding was a "happy coincidence of timing, luck, nepotism and Protestant work ethic."
This is a brief answer but hopefully gives you some insight. For better sources than I:
-John Lynch, An Unlikely Success Story: The Belfast Shipbuilding Industry, 1880-1935
-Sean McMahon, A Short History of Ireland
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