Kay,
The 1858 marriage certificate should give you the Graham’s townland (address). (Griffiths Valuation for 1862 lists a George Graham in Aghaleague and another in Tullanaginn. I don’t know if either of those is the address on the marriage certificate but if it is, then we can pinpoint where the family lived). GRONI keeps the statutory birth, death and marriage records which start in 1864 for births & deaths and 1845 for non RC marriages. Prior to that you need to search church records which are held in PRONI (the public record office) in Belfast. You don’t say what denomination the family were but I’d guess they were probably Church of Ireland. Magheraculmoney Church of Ireland parish records start in 1767. So you might be in luck. The church records at PRONI are not on-line and a personal visit is required to access them. Those records may have George’s baptism, plus that of the earlier generations.
I looked for the surnames Allat and Cannon in Fermanagh in Griffiths Valuation. There were none of either name. Likewise in the 1901 census of Fermanagh there were no Allats listed and just 1 household with Cannon. So not common Fermanagh names.
Magheraculmoney is in Irvinestown civil registration area. I searched for Margaret Graham deaths registered there from 1864 to 1908 and found just 3: 30.6.1872 aged 80, 7.1.1873 aged 76 and 24.5.1899 aged 96. You might want to look at those to see if any are George’s Graham’s wife. If his wife died before 1864 then you would need to search the burial records in Magherculmoney parish records (in PRONI). Not every burial was recorded though. It’s a bit hit and miss.
The reason why the family left Ireland in the 1860s was almost certainly the same reason people migrate today – economic betterment. Ireland has very few natural resources (no oil, coal, iron ore etc) and so did not benefit from the industrial revolution in the 1800s, the way Scotland, England, the US, Canada & Australia did, which created hundreds of thousands of comparatively well-paid new jobs in new industries (coal mining, steel making, railways, ship building etc). So that was a big pull factor. There had also been a huge population explosion in Ireland going up from about 3 million people in 1750 to 8 million in 1830. There simply weren’t jobs for all those people. In much of Ireland the only employment was subsistence farming topped up in Ulster and one or two other areas with a bit of linen weaving. And then the straw that broke the camel’s back, along came famine, numerous times throughout the 1800s. The worst period was when the potato crop failed almost completely 3 years in a row in the late 1840s, and then partially several more years after that.
People with small amounts of land were often very much one crop dependent, growing only potatoes because you could grow more potatoes to the acre than any other crop, and they need the minimum of maintenance, but as a consequence they had nothing else to fall back on when the blight attacked them, and because it was largely a barter economy they mostly had no spare cash to buy food. When the crop failed 3 years in a row, people ended up eating their seed potatoes, leaving them nothing to plant the next spring. It is estimated that during the years 1845 to 1850, around 800,000 people died of starvation or of a famine-related disease such as typhus, dysentery, scurvy or pellagra. A further two million people emigrated. Unlike earlier famines, in which the population recovers quickly from the catastrophe and continues to grow, the after effects of the Great Irish Famine were such that the population of Ireland, standing at 8.2 million people in 1841, declined to 6.6 million in 1851. Fifty years later, Ireland's population was still showing a decline (down to 4.5 million), even though every other European country was showing a population increase. Ireland’s population did not return to its pre-famine heights until 1964. Approximately 8 million people left Ireland between 1801 and 1900 - the equivalent of the entire pre-Famine population. The population today is only around 6 million.
Other factors led to the continued emigration too, eg early mechanisation on farms. With new machines to turn the soil and plant seed, farmers no longer needed an army of agricultural labourers to help on the farm. So those jobs were rapidly disappearing. Likewise mechanisation had led to linen factories being set up in places like Belfast. These made home weaving uneconomic and so also upset the labourer’s family economy. Agriculture was the biggest single employer in Ireland, but it was mostly a barter economy. Few people had any ready cash save what they could make from weaving or any government sponsored work such as building new roads. So when the opportunity arose to get jobs with a regular wage packet, as opposed to a few pence from your father each week, the decision to migrate wasn’t really all that hard to make. So it was as much about economic betterment as anything.
There are no local migration records as such in Ireland, just the passenger lists where they have survived. (No official body in Ireland or Britain kept records of who left or why they left. There was no reason to gather that information).