You are actually asking two very different questions: What started the Troubles, and if there was Unionist v Nationalist conflict during the height of the Shipbuilding era. If I have these wrong then please let me know.

There have been several great posts on the topic of past conflicts, so if you are interested in that question please see the following:

/u/girlscout-cookies on a brief history of conflict.

Thread on the ongoing tensions, has several good comments.

As you specifically mention the Titanic and the shipbuilding industry, I am going to cover the early history of conflict between Catholics/Nationalists and Protestants/Unionists. This should give you an idea of the early history of conflict that eventually evolved into the Troubles in 1968.

TL/DR: Yes, there were issues prior to the Troubles, particularly within Belfast and the shipbuilding industry.

In the 1830s there was a growing anti-Union sentiment among Catholics that followed on the heels of the 1829 Catholic Emancipation Act. Unionism, like Republicanism, strengthened in the 1830s and culminated in the first anti-Catholic riot in 1835 when the Minister Henry Cooke led a protest against Catholics living in the Sandy Row housing community located only a few blocks from the shipyards. Rioting subsided for the next two decades but tension did not, particularly in the 1840s during the Great Famine of 1845-47. As the Famine ravaged farms throughout Ireland, thousands of desolate families were forced to migrate from their ancestral homes, and for those that could not afford passage to the United States or England, Belfast offered hope. During this period of modernization and Famine, Belfast grew more rapidly than any other major city in the United Kingdom. To contend with the growing population, Belfast developed heavily segregated districts that were dispersed throughout the city. It is along these district lines that many of the earliest riots took place.

In 1864 the first major riot occurred between Catholics and Protestants. The riot lasted several days and affected various industries. Catholics were persecuted by Protestants while attempting a funeral procession from the Catholic Falls Road through the Protestant Sandy Row where the Friar’s Bush graveyard was. The riot spread to local linen mills and various factories where Protestants forcibly banned Catholics from work, despite the protestations of many employers. It was in the shipyards that the riot eventually broke: work in the shipyards was primarily reserved for skilled Protestant labor while Catholics made up roughly 1.5% of the workforce working unskilled jobs. With work slowing down several shipyard owners threatened to close the yards until Catholics were allowed back to work. The riot left 327 dead and further riots occurred almost yearly. The life experiences, ideals, beliefs, and social conditioning only act to perpetuate the growing Catholic and Protestant feud as housing, though segregated, was spread throughout the city. Had the city been segregated along clearer lines, to say that the Catholic housing districts were not as sporadically placed throughout the city, then the systematic rioting between groups over housing proximity would have dissipated instead of devolving into strict sectarian modes of thought.

There was a great deal more that happened over the following decades but let's jump to the twentieth century. Note that the Titanic was ordered in 1908 and construction was between 1909 - 1912.

By 1901 the population of Belfast reached 350,000 with Catholics making up around 24% of the populace. Shipbuilding continued to thrive in the early years of the century, as Workman & Clark increased their employment from 5,000 in 1902 to 9,000 in 1909. Despite the relative peace of Belfast, the issue of Home Rule for many Protestants, especially within the Orange Order—a Protestant organization that celebrated the 1680 defeat of King James II by William of Orange at the Battle of the Boyne—was still a source of tension. The Orange Order was accused by some of its own members that the Order no longer cared for the working class. This attack came took center-stage in 1907 when unionized Catholics and Protestants united in a strike together for better wages.

The National Union of Dock Labourers hired James Larkin to organize on the docks and shipyards. By April he had recruited 2,900 dockers and began negotiating better pay. The strike began shortly after negotiations led to the dismissal of several workers on a coal yard with scab labor brought in as replacements. The protest was successful in getting the union workers their jobs back on the yard. Larkin was recruiting from all levels of skilled workers with no restrictions on religion, which terrified the employers along the docks and shipyards so the Belfast Steamship Company instituted a union lock-out in an attempt to stem the growing support for Larkin. The lockout held and in June in an act of solidarity dockers, carters, and shipbuilders left their jobs. Over the course of the summer only about 3,500 workers were directly affected by the strike, but mass meetings, picket lines, and violence erupted involving between 5,000 and 10,000 supporters. In early July coal-workers joined the strike and several police officers assigned to protect scab workers mutinied and joined the protests.

Amongst the protesters there were no issues of sectarian conflict, though for some of their supporters that was far from true. On the July 12 Orange Day celebrations, which commemorated the defeat of James II, the Orange Order was split between supporters of the strike and those that refused to take part while Larkin was in charge. Supporters within the Order formed the Independent Orange Order and collections were made to benefit the strikers. The Independent Orange Order supported Larkin, a Catholic, on the grounds that sectarian division only hurt the workers, whom “would not be successful, because men of all creeds were determined to stand together in fighting the common enemy, the employer who denied the right of the workers to a fair wage.” For the old Orange Order, fear clouded judgment and Larkin was perceived to only be protesting in support of Home Rule and Nationalism. In August, this fear led to shootings within Catholic districts, the worst of which was by a soldier that killed two workers in the Falls Road, the second most concentrated Catholic district. For Catholics these acts of violence, even the shooting in Falls Road, did little to towards ending the strike. In September terms were agreed upon and the strike ended; however, even though the union declared victory the primary concession for returning all original union employees to work was that they would have to work with non-union workers and not earn a pay increase.

Even with the general support for the non-partisan strike, the unions lost power and sectarianism on a whole returned to a previous level. It is possible that the alliance of 1907 failed to become a mainstay of Belfast because in the case of Ulster city religious divisions ran too deeply to establish anything more than a limited economic unity. Not until July 1912 Home Rule riots was there a major incident between Catholics and Protestants, which is why we can assume that given the sectarian nature of Belfast and her populace civic peace could only be attained when the issue of Home Rule was no longer a concern. Through the second half of the nineteenth century riots and disorder intensified, culminating in the Home Rule riots of 1886 and 1893. From 1893 until 1907 the conflict between the partisan groups of Belfast eased as Home Rule had lost all power as a possible legislative topic in Parliament. Until 1912 Home Rule was no longer seen as an eminent threat, resulting in years of relative peace amongst the Belfast populace.

I know that this is way too long but hopefully you now understand a bit of the history behind the Troubles, especially through the construction of the Titanic.

Sources:

J.W. Boyle, “The Belfast Association and the Independent Orange Order, 1901-10,” Irish Historical Studies, 13 no. 50 (September 1962).

John Lynch, Forgotten Shipbuilders of Belfast: Workman, Clark, 1880-1935 (Belfast: Friar’s Bush Press, 2004).

Ronald Munck, “Class and Religion in Belfast-A Historical Perspective," Journal of Contemporary History, 20 no. 2, (April 1985).

Anthony Hepburn, A Past Apart: Studies in the History of Catholic Belfast, 1850-1950 (Belfast: Ulster Historical Foundation, 1996).

Michael Moss, Shipbuilders to the World: 125 Years of Harland and Wolff, Belfast 1861-1986 (Belfast: Blackstaff Press, 1986).