DD is a baptized Catholic (as are DH and I), and she was 6 months old at her ceremony.
We had to pick 2 godparents, so we chose my brother and DH's sister. We asked them to be the godparents shortly after we decided to baptize DD, which was maybe a month or two after she was born.
If you want to baptize LO at 1-2 months, I would say that you should ask your intended godparents soon.
BTW, I think that age is fine, but I personally felt like a big old mess after DD was born and probably wouldn't have been able to get myself together and looking presentable until closer to 2 months. Everyone is different though.