Really I can't provide you with additional hard information. Instead here's the current version of what I "know". I'm continuing to research through relatives in Northern Ireland.
Patrick, my great-grandfather (born c.1830) was known as Patrick Montgomery. His son was known as William Hugh Montgomery at birth, though when he married and later died his name is recorded as William Hugh McGivern. That William Hugh married Elizabeth Walsh around 1903 we believe.
Oldest living relative in Ireland (another d William Hugh McGivern, son of William Hugh and Elizabeth, and aged 88!) says that Patrick's father was also called William Hugh, and his (Patrick's) grandfather was called, like him, Patrick.Except while Patrick styled himself Montgomery, Patrick's grandfather was called McGivern. We don't know whether William Hugh (Patrick's father) was called McGivern or Montgomery.
There is an amusing family folk-tale about how that William Hugh arrived to fight at the Battle of Ballynahinch (a significant episode in N. Irish history)the day after the Battle took place.
That battle took place in 1798(?). However, it was essentially a Protestant/ Presbeyterian uprising. Montgomery is an Ulster planter names whereas McGivern is a Ulster gaelic name. There was apparently very limited Catholic gaelic participation in the uprising of 98 in Down. Most likely if William Hugh was an (intending!) participant in that battle he was a Protestant/ Presbyterian. So he may have called himself Montgomery, since at that point at least, family names were easily identified with particular religious affiliations.
One need's to be extremely careful however, about these inferences. Most people don't realise that the relationships (cultural, social and religious) between the Ulster planters/ Ulster Scots and the Ulster Gaels was very much more complex and dynamic than the popular view allows ! (For example there are many New World McGiverns from that time who have, or quickly established, a Presbyterian heritage. And it is also well known that families changed names for reasons of expediency as well as changed religious convictions.)
There is evidence too that William Hugh styled himself as McGivern in his own lifetime. Apparently, in the turn of the 19th century Land Records of the Downshire family he is known as William Hugh McGivern.
Is that William Hugh related to your family? Well if there is a Montgomery link in your background, maybe. If not, maybe, maybe not. There were numerous McGivern's in the Counties of Armagh and Down ( though mostly in South Down)during the last century. My casual observation is that they had a great fondness for names like William, William Hugh, Patrick, Daniel, Thomas and John ! A fondness they brought to the New World with them ! So I don't think the names alone are other than suggestive.
I'm sorry that I don't have more for you. My particular focus is to understand more about the peculiar interchangeability of the names McGivern and Montgomery in North Down.
The difficulties of researching this are compounded by the fact that many official records were apparently lost in a fire in the Dublin Public Records office in 1916.
However, numerous other historical records (parish registeries, rent records, tax records) survive but most are not easily accessible, nor are they complete or accurate. And most provide only indirect or supporting evidence.
I intend to post harder information on this site and on the Montgomery site, when I have assembled everything I can. I expect to do this within the next few weeks.