Statutory birth, death and marriage registration (in some jurisdictions called Vital Records) only started in Ireland in 1864, save for non RC marriages which were recorded from 1845 onwards. So you probably won’t find statutory birth, death or marriage certificates in Ireland for this family. For earlier years you usually need to rely on church records, where they exist. You obviously need to know the precise denomination in order to search the correct records. Not all churches have records for that period and not all are on-line.
RC records are mostly on-line on the nli site:
For other denominations, the churches usually hold the originals but there are also copies in PRONI, the public record office, in Belfast. A personal visit is required to access them. Access to the records there is free. This link explains what records exist, parish by parish:
If you are unable to go yourself, you could employ a researcher. Researchers in the PRONI area: http://sgni.net
If Francis’ father was a farmer then, as I am sure has already occurred to you, he’s unlikely to have been born in Belfast itself, since you don’t find many farms in city centres. He could have been born nearby as you say. Belfast is mostly in Co Antrim with a smaller part in Co Down. Griffiths Valuation c 1864, lists most farmers in Ireland at that time. Unfortunately James Robinson is very common name and there are about 40 listed for those 2 counties. If you know Francis’ denomination you could get a researcher to go through the church baptism & marriage records for each of the parishes where there was a James Robinson in Griffiths. But that'd be quite a lot of work. Marriage and baptism may not be in the same church. Tradition was to marry in the bride’s after which she’d normally attend her husband’s.
I searched the rootsireland site for Francis’s baptism and for his parents marriage but without success. However as I mentioned before a lot of records – especially protestant ones – are not on-line.
Assuming James & Ann were both alive in 1864 then there should be a death certificate for them in Ireland. I searched deaths for James 1864 – 1901. There were 449. 288 named Ann. A bit of a needle in a haystack, I think. Irish death certificates don’t list a woman’s maiden name (at least they didn’t in the 1800s) so no clues there.
Sorry I can’t provide you with more positive suggestions.