Is it mentioned anywhere in all of the Mishna or the Talmud that Rabban Gamliel converted to Christianity?
We can assume he did not.
Although it would be hard to prove a negative, the fact of the matter is that the talmudic rabbis never shied away from mentioning when a great and respected religious leader left the fold. See for instance about Yochanan Kohen Gadol and Elisha Ben Abuya.
In a similar vein, mention is also made of Rabbis who lost their minds such as Ben Zoma in that second link, or Rabi Yochanan seen here (bottom of 84a) leading us to assume that the other rabbis in the talmud did not.
So we can safely claim that Raban Gamliel did not become an apostate or lose his mind.
No. This is purely and completely untrue.