Did the Silence want to stop the events of "The Name of the Doctor?"

The Silence didn't care about the Doctor visiting his tomb. The prophecy is about the battle that resulted in the graveyard we see in The Name of the Doctor. During that battle, Silence fell. The main reason I think this is the case, is because:

when no living creature may speak falsely or fail to give answer

wasn't really part of the events of that episode. The Doctor clearly failed to give an answer, River was the one who provided the answer.

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We see the events of the prophesy play out in the recent Christmas Special episode, The Time of the Doctor. In it, The Doctor and Clara go to a world revealed to be Trenzalore. The crack from Series 5 is in a building on the planet, in a small town. It is emanating a 'truth field', which prevents people from 'speaking falsely' (see above prophecy). A signal is coming from the crack, which contains an encrypted message asking a question over and over:

Doctor Who?

They come to the planet, and find many ships from various races. Each race sent one ship to investigate the signal, which is being transmitted throughout time and space. The Church got there first. To protect the planet, they setup a shield to prevent anyone from going to the planet or attacking it. When the Doctor arrived, The Church sent him down to investigate.

The signal is coming from Gallifrey, outside the universe after the events of Day of the Doctor. If the Doctor speaks his name they'll return, but in so doing, will restart the Time War. Upon learning this, The Church dedicates themselves to Silence, the silence of the Doctor, as they do not want the Time War to return. As The Church's leader proclaims:

Silence must fall.

A repeating theme of the 11th Doctor's run, and something related to the prophecy, "Silence will fall when the Question is asked" (Let's Kill Hitler).

So The Church helps maintain a truce, they help the Doctor prevent any of the races from invading the planet, and he does not answer the question. Then time passes as they maintain the truce. Centuries pass, the Doctor grows old.

At the end of the episode, the Daleks break the truce and invade the planet. Clara argues into the crack that the Time Lords don't need to hear the Doctor's true name, they just need to help him. They do so by providing him with a new set of regenerations. The 11th Doctor dies of old age, blasting the Daleks with his regeneration energy, saving the planet as his regeneration begins.

And so the Eleventh falls. This fulfills the prophecy, and explains, "what Silence?" It's the Doctor's silence; his refusal to answer the question. This gets copied into Church faith, which ends up being the source of the past events where the Silence try to kill the Doctor.

The Silence are a faction of The Church who took the new Silence dogma and went back along the Doctor's personal timeline. They attacked him earlier (i.e. by creating River Song, by blowing up the TARDIS) to kill him, preventing him from reaching Trenzalore. Essentially they sought to ensure that 'Silence would fall' by preventing the Doctor from being present.

Largely copied from my answer over on Movies & TV.SE