If Joshua was born in 1859, then that’s before the start of statutory birth registration in Ireland (1864) so you won’t get a birth certificate for him. Prior to 1864 you need to search for baptism details. To do that you need to know his denomination. Carrickfergus has a number of churches: Congregational (Baptisms start 1819) Church of Ireland (1740), Methodist 1815, 1st Presbyterian 1823, 2nd Presbyterian 1852, Loughmorne Presbyterian 1848, Woodburn Presbyterian 1863, RC 1852. Copies of all those records are held in PRONI in Belfast. In addition, the RC records are on-line on the NLI site.
If you can’t get to PRONI you might need to employ a researcher to go there and go through the records for you.