It is quite possible that your family moved from Scotland to Ireland in the 1600s. Something like 200,000 Scots made that same move during that century. Some as a part of the Plantation of Ireland and others because of famine in Scotland in the 1690s. However there are no records of movements between Scotland and Ireland in those days (or even now). It was, and still is, just a short domestic journey.
Tracing families in the 1700s in Ireland can be difficult. Prior to statutory registration in 1864, the main records in Ireland are church records but not every parish has any for the 1700s. You haven’t said what denomination the Alexander family was but if they came from Scotland then it’s possible they were Presbyterian. There are many Presbyterian churches in Co Armagh (reflecting the fact that large numbers of Scots settled there) but only a few have records for the 1700s. The only Presbyterian church in Co. Armagh with records for the early 1700s is Armagh (city) which start in 1707. You might want to search them.
I don't think they are on-line anywhere but there is a copy in PRONI (the public record office) in Belfast. You need to go in person to view them. If you can’t get there yourself you might need to hire a researcher, or PRONI may search for you (for a fee).
Researchers in the PRONI area: http://sgni.net
Contacting PRONI by e-mail, letter or phone:
PRONI
2 Titanic Boulevard
Titanic Quarter
Belfast
BT3 9HQ
Phone: 028 9053 4800
Fax: 028 9025 5999
Email: proni@communities-ni.gov.uk
