I am not having much luck.
In the Derrynoose registers, they have children:
Patrick Morgan baptised at Derrynoose on the 1st of February 1827. Parents Alexander Morgan and Nancy Lennon.
Sponsors were Bernard Morgan and Rose McGeough.
I haven't found further information on Patrick.
John baptised at Derrynoose on the 30 June 1829.
Parents Alexander MORGAN and Ann LENNON.
Sponsors were James LENNON and Mary MORGAN.
Bridget Morgan baptised at Derrynoose on the 1st of Feb 1833.
Parents Alexander Morgan and Ann Lennon.
Sponsors were Bernard Barka and Ann Barka.
Ann Morgan baptised on the 8th of December 1846.
Parents Alick Morgan and Nancy Lennon.
Sponsors were Ann and Francis Morgan.
Baptisms 01 Feb. 1835 to 29 Jan. 1837
Baptisms Dec. 1846 to 28 Jan. 1866
Marriages 17 July 1846 to 31 Jan. 1875
Deaths 22 July 1846 to Apr. 1851
and 77 images on Microfilm 05589 / 04 for
Baptisms 01 Feb. 1866 to Feb. 1881
Marriages 08 Feb. 1875 to 16 Jan. 1881
I found one Alexander Morgan at Rowan, Derrynoose in the Griffiths Valuation for Ireland leasing house, offices and land from Andrew J Crawford. No results found in Tithe Applotment books for either Armagh or Monaghan.
I found a death record for an Alexander Morgan who died on the 24th of September 1870 in the Parish of Keady. His age was 75, his address was Roughan, occupation farmer. Informant was a householder by the name of Catherine Murphy. This could fit my 3rd great grandfather.
Also in that Parish was the death of an Agness Morgan on the 28th of December 1869 age 65, widow of a farmer, address Fergort, information a householder named Francis Morgan. But as this Agnes was a widow at the time of her death she can't be the wife of Alexander Morgan.
Valerie Prince, a very helpful member of the County Armagh Genealogy group on Facebook searched in Armagh and Monaghan for an Ann Lennon's baptism, 1804 +/-5 years.
Ann Lennan baptisms in 1802, 1803, 1804, 1806, Armagh.
A couple of trees in ancestry have Ann Morgan nee Lennon's parents listed as Felix Morgan and Mary McArdle but none of these baptisms have a father named Felix.
Elaine Curran, another very helpful member of the group noted that a Miles Morgan and Ann Lennon married at Derrynoose in 1829.
I found that Miles and Ann were having children at the same time as my Alexander and Ann. The ancestry trees naming Ann's parents as Felix Lennon and Mary McArdle have listed these exact children for Alexander Morgan and Ann nee Lennon.
Francis baptised 2/12/1832
Thomas baptised 24/12/1835
Michael baptised 20/01/1847.
So I think that confirms that they aren't children of Alexander and Ann and I am still stuck on finding parents for Alexander and Ann.