There is only 1 person listed in the telephone book for Sligo Town, Sligo spelled Connellan. Address sufficient for mail delivery.
36 Conlons in Sligo, too many to write to. You could start with the spelling of Connellan above. You can also just write to the males listed as Conlon perhaps 10 at a time and see
Most genealogy websites with parish records don't stretch to 1937 usually ending in 1900. The 1901/11 census won't help you either.www.familysearch.org
has civil records from 1864-1950's
No Connellan but other spellings similar
Patrick J. Conlon civil registration Jan-March, 1936
There are others but this is the closest 2 +/- years.
You maybe able to get a copy of his birth certificate
from the General Registry Office www.groireland.ie
but usually it helps to have parents names, however, they will do a search around the right date.
Hopefully the bell will ring with someone related