Statutory recording of births and deaths only started in Ireland in 1864, so this may be why you cannot find any births before 1866. Prior to that you need to rely on church records (where they survive). In this case, the following records survive for Killinkere parish. A copy is held in PRONI, Belfast. (The records are not on-line and so if you are not able to visit yourself you might need a researcher).
C.I. Killenkere (Kilmore diocese) Baptisms, 1840-44 and 1878-99; marriages, 1840-42, 1845-74 and 1877-98; burials, 1877-1900; Sustentation Fund account books, 1883-1900; parochial accounts, 1872-1901; vestry minutes, 1826-70.
I wonder if this might be your family in the 1901 census. Some of the names and ages match, as does the religious denomination:
Griffiths Valuation for 1855 has a James Mahood farming on Freeduff (which is in the parish of Drumlumman). He had a house, offices (outbuildings) and 33 acres of land (plot 5). Just off the modern R394 road beside the old railway line to Cavan town.