Beverly,
A birth in 1837 or thereabouts is long before the start of statutory birth registration in Ireland (1864) so you won’t get a birth certificate for Alexander Dunn. You may find his baptism but to do that you need to know his religious denomination. I looked at the 1901 census of Co Tyrone. There were 197 Dunns. The majority were either Church of Ireland or Presbyterian. Neither the Church of Ireland not the Presbyterian church in Pomeroy have any records for the 1830s. The Church of Ireland lost theirs in the 1922 fire and the Presbyterian church’s records only start in 1841. Prior to that they either weren’t keeping records or they too are lost.
As far as Alexander’s own date of birth is concerned, most people in Ireland didn’t celebrate birthdays in the 1800s, and rarely knew when they were born. If officialdom later asked for an age or date of birth it was common just to make one up. So the normal advice is search 5 years either side, though unfortunately in this case that probably still won’t help. If he was Presbyterian, you might find any younger siblings in the Pomeroy records. You really need to get someone to go to PRONI in Belfast to look them up as they are not on-line anywhere.
I looked in the tithe applotment records for Pomeroy in 1829. There was only 1 Dunn farming in the parish and he was named Thomas so I’d say that’s a very good chance it’s Alexander’s father:
http://www.irishgenealogyhub.com/tyrone/tithe-applotment-books/parish-of-pomeroy.php
He was farming in Cappog (more commonly Cappagh). He had gone by the time of Griffiths Valuation in 1860 suggesting he had died. However there was an Anne Dunn farming there on plot 22 (a 20 acre farm). A woman farming was usually a widow, so I’d hazard a guess that might be Alexander’s mother. Do you have her name from Alexander's death certificate perhaps? It’d be a way of confirming.
Plot 22 today is up a lane on the modern Sessiadonaghy Rd. The original farmhouse has been demolished and new buildings erected but the land is still farmland.
https://www.askaboutireland.ie/griffith-valuation/index.xml?action=name…
Anne Dunn died in 1877 aged 62.
https://civilrecords.irishgenealogy.ie/churchrecords/images/deaths_returns/deaths_1877/020542/7208386.pdf
After her death, the Valuation Revision records tell us the farm was split. Patrick McKeown took some of the land and a James Dunn senior took the rest. He is deleted in 1896 and there are no Dunns farming there by the 1901 census. I don’t know what became of James Dunn after 1896.
