On the subscription site RootsIreland, I found the following:

Alexander McMurdie

Baptism:26-Feb-1716

Parish/District: ARMAGH 1

Co. Armagh

Presbyterian

Father: Gauin [likely Gavin] Mcmurdie

I cannot swear that this is your Alex, but the name, age, and place seem to match the Alex you are seeking.  By any chance did your Alex name his first son Gavin? That would be another clue.

The above Alexander had siblings (listed with baptism date): David 1713, Arthur 1720, Elenor 1723, Elenor 1726.

Thank you so much for this information, Patricia! This Alexander definitely does look the right age, religion and birth location. However, the name of Alexander's father and all of his siblings do not match any of his children or grandchildren.

My Alexander's children were Ann 1750, George 1753, Alexander 1755, Bridgit 1759 Hugh 1760. His wife's name unknown. 

Continuing on... Alexander 1755 married Mary; their children were Charles 1773, James 1774, Henry 1775, Catherine 1780 and Alexander 1790. 

Based on these 3 generations, it looks like this Alexander McMurtrie family line did not follow traditional Irish naming patters opening up the good possibility that Gavin could be his father. 

One other clue to verify accuracy is... If Alexander (above) was the right one, He would have come to America when he was 7, so he had to come with his parents. Do you have any evident of Gavin and his children living in Ireland past 1723?  

Also FYI, I have a pretty knowledgeable ancestry friend who suspects that Alexander could be possibly one of the Scottish Protestant settlers who had been sent in to Ireland to colonize the area after the Cromwell incursion. Do you know anything about that? 

Thank you for your time and effort, Patricia. 

What your friend is referring to is the "Plantation of Ulster", which actually began in 1606, well before Cromwell's time, though it continued for a century or so, through the Cromwellian period.  Wikipedia has a good article about the Plantation at this link:

https://en.wikipedia.org/wiki/Plantation_of_Ulster

It's very likely that your McMurtry's were Scottish settlers who came to Ireland at that time.  The Gaelic surname Mac Muircheartaigh ("son of the navigator") originated separately in Ireland and Scotland.  It was anglicized in various ways.  The forms MacMurtrie, MacMurtry, and MacMurty were commonly used in Scotland, and brought to Ulster in those forms during the Plantation, principally to County Antrim, but another anglicized form, MacBrearty, was found in County Donegal.  The name was also sometimes anglicized as Murdoch or Murdock.

In Ireland, the Mac Muircheartaigh (same meaning in Irish Gaelic as in Scottish Gaelic) surname originated outside of Ulster, and was usually anglicized as Murtagh, Murtaugh, or Murtha, but also sometimes as Murdoch or even McMurtry.   A separate Gaelic surname, Ó Muircheartaigh ("descendant of the navigator") also existed in Ireland, and was usually anglicized in one of those same ways, but also sometimes as O'Moriarty.

Thank you, Kevin. Very interesting information. What kind of documentation is available during this time period? 

Your family being Presbyterian is a strong clue that they are of Scots origins. Presbyterianism was established in Scotland in the mid 1500s and brought to Ireland by Scots settlers.  You don’t find many people of “native Irish’ background who were Presbyterian. Most of the Scots settlers went to the counties of Ulster, which includes Armagh. So you have a Scottish surname, Presbyterianism and residence in Ulster. That’s 3 pointers to tell you that the family are almost certainly of Scottish origins.

There were several waves of migration to Ireland throughout the 1600s as Kevin has explained. There was big rush in the 1690s due to famine in Scotland, and also because of the Killing Times in the 1670s and 1680s when many Presbyterians fled Scotland.

If you want to learn about that period in more detail, a good read, is Dr David Hume’s book: “Eagles Wings, - the Story of the Ulster-Scots and Scotch – Irish.” It explains why they came to Ireland, broadly where they came from in Scotland and why after a short time in Ireland many decided to move on again to North America.

The only record I found indicating Gavin was in Ireland after 1823 was the baptism of his second daughter Elinor in 1826. I assume he was in Ireland then.

Very interesting info, Elwyn!  Thank you. 

And thanks again, Patricia. I think I should check for their records in Scotland now. Appreciate all of you and your help!  Maureen