The fine print there says that "the Baptist Churches declared nonfellowship with the Irregular Churches [ie. nonbaptists]" in 251 AD.
Does anyone know what that could possibly be referring to?
I have no idea. I also don't know who the "Paterins" were; a Google search turns up only non-English results.
As far as I know, the only church in the 1st millennium to reject infant baptism* was a faction with the Nestorian church that also accepted Pelagianism (Pelagianism and Nestorianism having been condemned at the same council). This faction argued that since infants are without sin, it makes no sense to baptize them (since baptism is for the remission of sins). Needless to say, this was a very different reason than that given by modern Anabaptists and Baptists.
There were also some people in the mainstream Church who delayed baptism as a tactic
(based on the idea that sins committed before baptism did not need to be repented of, and thus it was best to be baptized as late as possible, especially on the deathbed). The Church Fathers were highly critical of this attitude.
*Apart from, of course, dualist and Gnostic sects who rejected baptism completely.