Hi Johnathan,
The Hamburg passenger list shows Friedrch Bertram and his wife Anna arriving 1863 in Australia - and their previous residence in Germany was Mechtshausen in Hannover.
Mechsthausen belonged to the Catholic parish of Bilderlahe in Bistums Hildesheim. The church books are available online free of charge at Matricula.
The christening record for Andreas Christian Friedrich Bertram on 12 November 1837 shows he was born on 30 October 1837 to Christian Bertram (might be Bartram), Tagelöhner from Gadersheim, and Wilhelmine Röttger. Here is the link (it is entry nr. 9 on the page): Taufe - 0145_01 | Bilderlahe, St. Michael | Hildesheim, rk. Bistum | Deutschland | Matricula Online
Hope this helps,
Nancy