Hi Mary, the fact that you know his father's name is hugely helpful as there are likely to be more records relating to his father than to him as he emigrated when possibly quite young.
Searching the tithe records (indexed on www.rootsireland.ie for a number of counties), I found two individuals named George Brady renting land in Co. Cavan in the late 1820s/early 1830s. This is not a full listing of all those of that name as it only listed those who were renting land not those renting only a house, for example in a town or village or non-farmers in the countryside.
George Brady, Cullis townland, Annagelliff parish, Co. Cavan, 1828
George Brady, Tonymore townland, Kilmore parish, Co. Cavan, 1831
The website www.rootsireland.ie does not yet have records from every church on its site as this is an ongoing project. Once you have explored everything available online, we offer a wide range of research services and would be delighted to visit the archives and search through the church registers not yet online looking for George's baptism. You can find out more here: https://www.ancestryireland.com/research-services/
All the best with your research,
Ulster Historical Foundation