The second reference listed in the question,

Charles Frohman and Benjamin Fine, "Some Amalgam Structures for Bianchi Groups," 1988, Proceedings of the American Mathematical Society, Vol. 102, No. 2, pp. 221-229,

does in fact include a proof of the statement in question. The article is available here: arxiv.org/pdf/1204.6697.

The second reference listed in the question,

Charles Frohman and Benjamin Fine, "Some Amalgam Structures for Bianchi Groups," 1988, Proceedings of the American Mathematical Society, Vol. 102, No. 2, pp. 221-229,

does in fact include a proof of the statement in question. The article is available on Jstor.