The Doctor trusts River because she knows his name... He seems to assume that she knows his name because he must have trusted her enough to tell her at some point, but what if he didn't?
We already know that River can write Gallifreyan so it stands to reason that she can read it too. The Doctor's cot had writing on the end that the TARDIS didn't translate, but River saw it too. Perhaps that is the Doctor's name, River only knows it in her future because she read it there on his cot in her subjective past.

Perhaps River Song is just that good at acting, her entire past (or future) with the Doctor is just a ruse to get him to trust her so she can get close enough to kill him.

I know, wild fan theory, but it makes me think.