Infant Baptism in the Talmud

Today, I was in a room where Anticatholics, who are Reformed, in this case reformed Baptist, insisted infant baptism is completely without basis in ancient Israel, one of them even said Jewish Encyclopedia does not mention it, and he also alluded to the Talmud implying that it too does not teach infant baptism. I objected and said if a baby were part of the family of a "ger" a convert to Judaism--the child would be baptized. I showed them this website which states the procedures for a baby converting:

The conversion of a female infant or child according to Conservative and Orthodox practices only requires tevilah (immersion in a ritual bath called a mikveh). A male child also requires immersion in the mikveh. While parents might have trepidation about immersing a newborn completely in water, the supervising rabbi will facilitate the procedure with great care ensuring that the infant is safe during this momentary immersion.....A beit din, usually consisting of three rabbis, is convened for the immersion. Parents can enter the mikveh. If the children are old enough, they recite the needed prayers; if not, a rabbi does so for them. After the tevilah ceremony is completed, and a name chosen if one has not already been selected, the child is declared by the beit din to be Jewish.

Furthermore, I have found in the Talmud, which they requested I use, it states:

R. Jose stated, It happened at 'En Bol that the infant was made to undergo ritual immersion  before her mother; and Rabbi stated, It once happened at Beth She'arim that the infant was made to undergo ritual immersion  before her mother; and R. Joseph stated, It once happened at Pumbeditha that the infant was made to undergo ritual immersion  before her mother;  One can well understand the incidents spoken of by R. Joseph and Rabbi  since [immersion was necessary as a protection for] the terumah  of Palestine; but why was that necessary in the case spoken of by R. Joseph,  seeing that Samuel had laid down: The terumah of a country outside the Land of Israel is not forbidden unless [it came in contact] with a person whose uncleanness emanated from his body,  and this applies only to eating but not to contact?-- Babylonian Talmud: Tractate Niddah 32a

 As we see, this issue was not even one that Rabbis debated over, it was already accepted as a fact that babies were baptized, and why wouldn't they be, they can become ritually "unclean" like adults! 

What, however, of the following statement of Raba. 'If a pregnant gentile woman was converted, there is no need for her son to perform ritual immersion'.  Why is there no need for him to perform immersion?-- Babylonian Talmud: Tractate ‘Abodah Zarah57a

Note the discussion here is why SHOULDN'T a baby still in a mothers womb be subject to baptism after birth, the reason is because his mother while still pregnant converted(and assumedly was baptized) therefore making the child have no need for baptism in the Jewish mindset.

Also, Tractate Kethuboth11a discusses the conversion of minors, including ones less than "three years and one day old" and states "A minor proselyte is immersed by the direction of the court."

There are other sources too, but this should suffice for now.