If a character wants to provoke some sort of reaction (be it anger, or even just breaking his self-imposed silence) by talking to another [player] character; which skill is best suited for a skill check?
For social interactions between players, I tend to discourage the use of skill checks. This sounds like a strictly role playing situation. Instead of rolling a die, I would encourage the players to get in the heads of their PC's. Know why your character is trying to remain calm or giving the silent treatment. If the other person can rile you up enough, then get angry and start shouting. I wouldn't want a die roll to tell me how I react to another person at the table.
That said, there are certain times where it's somewhat difficult to role play your character (for example, if your PC has a lower wisdom than you do). In these cases my group usually has the player of the low-wisdom character set their own DC and make a modified wisdom check. On a success, they keep their cool and act as they themselves would. On a failure, they do the opposite.
But, if you have your heart set on using skill checks, I would suggest that either Intimidate or Diplomacy would work.
You can change another’s behavior with a successful check
You can change the attitudes of others with a successful Diplomacy check
The established rules allow for both of these skills to be used with different chances of success. If the player is very wise and highly resistant to fear, intimidate is probably a bad choice as they would likely win the opposed roll. Diplomacy may be your best bet since the player is indifferent or friendly to you, and the DC isn't too bad.
I'd turn to Intimidate here.
You can change another’s behavior with a successful check. Your Intimidate check is opposed by the target’s modified level check (1d20 + character level or Hit Dice + target’s Wisdom bonus [if any] + target’s modifiers on saves against fear). If you beat your target’s check result, you may treat the target as friendly, but only for the purpose of actions taken while it remains intimidated. (That is, the target retains its normal attitude, but will chat, advise, offer limited help, or advocate on your behalf while intimidated. See the Diplomacy skill, above, for additional details.) The effect lasts as long as the target remains in your presence, and for 1d6×10 minutes afterward. After this time, the target’s default attitude toward you shifts to unfriendly (or, if normally unfriendly, to hostile).
Emphasis added. Provoking a reaction that's largely against the will of the target seems to fall under its purview, even if the means of doing so is less hostile than typical.
I agree that using social skill checks for character interaction is unwise. I disagree with others about the skill checks that should be used.
The simple answer is that there is no mechanic in the Core pathfinder rules (as far as I can find) to provoke a character into attacking. Putting my DM hat on...
I would rule that Bluff is the most appropriate skill for provoking an enemy. In terms of fluff, this just makes the most sense. When you taunt, you are not trying to woo them (as with Diplomacy), and you are not trying to frighten them (as with Intimidate).
The goal is to trick them into doing something that they otherwise would not do. Think about the example of a swashbuckler, matching wits with a foe in a duel. The swashbuckler would throw witty, pointed remarks at an opponent to throw them off balance, and make them flub their attacks - I'd call that a Bluff check. I believe there is some class in 3.5 with a feature that supports this (it may even be Swashbuckler); but, I don't know the mechanics off of the top of my head. I've heard the term "called shot to his self-esteem" used.
You can force the character to break the self imposed silence by deeming the character to have said something, but you can not force the player to actually say anything in character. It is going to be hard to convince the player not to start a new self imposed silence. In addition interposing yourself into a feud between 2 other players is not going to work out for you. The fact that both players are there indicates a willingness to eventually work together again. If that did not exist then the player would simply stop coming to the games.
If you are the player trying to get the other person to talk to you read the last line above.