Terry,
The population of Belfast c 1710 was about 2500 people, so not very big. If your ancestors were born there, then sadly there are no baptism records for that period. The earliest church records for the town date to 1723 (Rosemary St, Presbyterian church). Their baptisms start 1723, marriages in 1741. You might just find the couple’s marriage. No harm in checking the Church of Ireland records too which start in 1745 (St Anne’s). The first RC church in the town didn’t open till the 1780s, so it can really be ruled out.
There are copies of the church records in PRONI (the public record office) in Belfast. Sadly there are no street directories or similar records for the early 1700s. And there are no records of the individual settlers who moved to Ireland from Scotland, England and Wales in the 1600s. Not a huge amount to go on, sadly.
You haven’t said what denomination your ancestors were but given the location (ie Belfast) and the era of their departure for North America (1700s) they were probably Presbyterian. Most migrants to north America in the 1700s were Presbyterian and at that time 90% of the population of Belfast was Presbyterian too. The town was established by Scots who moved to Ireland in the 1600s. There is however the possibility of English origins too. MacLysaght’s "The Surnames of Ireland" says when found in Ulster, Morrow is usually of English origins. So you should bear that in mind as a possibility.
Passenger lists weren’t routinely compiled for emigrants in the 1700s so tracing when someone left can be difficult. However there are some clues from correspondence from early migrants and also from the thank you letters they sometimes published in newspapers if they had had a reasonable crossing. Often most of the passengers would sign the letter. (They sort of acted a reference for the Captain when seeking passengers for his next voyage). Here’s an example, with a Morrow on board:
The following being the Passengers of the Ship CANTON, on a passage from Philadelphia to Ireland; beg leave to return Captain Hugh MaKee our sincere thanks for his kind and generous treatment to us, both as a Gentleman and experienced Commander on board the said Ship, and as we all intend returning to the Continent of America, he shall be our only choice, and well deserves our interest in the Passenger Line. Signed separately on board the said Ship, 7th day of February, 1792. James Abbit, Dublin John McGregon, do. Wm. Gallagher, Donegall Galb. Hamilton, Tyrone Geo. Mitchel, Down John McGonegon, Derry David Duff, Tyrone Robert Grimes, do James Willson, Antrim James Lesley, Antrim John Shields, do. Chas. Sweeney, Derry Thos Cook, Donegall Nathan Greg Cambel, George Masson, do. Antrim Robert Patterson, Down Jeremiah Barns, do. Andw. McClean, Armagh Wm. Mc Ferran, do. John Donnally, do. Job Buckson, Derry David Ray, Monaghan James McKee, Armagh John Rankan, Derry John Morrow, Derry Geo. Nevil, Dungannon James McMurray, do. John Culley, Down John Boyd Down, do. Peter Faggundeen, Henry McCann, do. Philadelphia, Robert Stewart, Antrim And. Turnston, Tyrone Thos. Price, Down Edw. Brein, Tyrone Fran. Osburn, Armagh Charles Gray, Donegall Arch. Cooney, do. Wm. Young, Antrim Henry McCrab, Derry
These sorts of records can be found on the Irish Emigration Database (IED).
http://www.dippam.ac.uk/ied
I searched for Morrow 1700 – 1760 but sadly did not find any.