Hi - I put John Davison b.1838 in www.rootsireland.ie [Irish Family History Foundation] "Search for your Ancestors" and they say they have 15 possible bapt/birth records for him, and six possible ones for John Davison b.1813 (plus or minus 5 years for both).
Also 4 possible marriage records for John Davison b.1813. As you know his wife's forename this could be the best bet.
This website used to be free to search, and they still send me new records added, but now "You must purchase a subscription to view the search results". (US$29 for a 1 month subscription.)
It might be worth a try in order to get his marriage details, and if you are lucky, you could then look at a lot of records for both of them during the rest of that month. (Using variations for the spelling Davison/Davidson.)
See also the National Library of Ireland website: www.nli.ie/en/griffiths-valuation.aspx
"The primary valuation of Ireland or Griffith's Valuation - carried out between 1848 and 1864 to determine liability to pay the Poor rate (for the support of the poor and destitute within each Poor Law Union) - provides detailed information on where people lived in mid-nineteenth century Ireland and the property they possessed. Griffith's Valuation is fully searchable online, free of charge.." www.askaboutireland.ie/griffith-valuation/