While I like the general idea behind the new PvP quests since they actually are more PvP focused than previously, they are horrible when looking through the lens of influence pushing. You now get 1-2 quests which can be completed if the enemy faction is not coming to fight. This dramatically decreases the speed at which you can put a territory into contested status. Additionally, the 5 kill quest is extremely flawed as you must kill 5 people without dying and return to town. This favors zerg style play as if you are going into equal fights the probability of getting 5 kills without dying will be pretty low (this is not accounting for if you are far superior in skill or numbers than the other players).
Please return the 3 boring quests as the standard quests for influence pushing and add the more PvP focused quests as additional quests which can be completed rather than having the cap at 3 quests. In the past week on my server there were 2-3 wars and while some can be attributed to the holidays, the change to the influence quests has definitely been a major factor.