Rather than a single incident, it may have been several:
Strike 1: The first contact
The first contact was depicted in ENT 'Broken Bow' and is described by Memory Alpha as follows:
After arriving on Qo'noS, Archer, Klaang, Sato and T'Pol enter the Klingon High Council Chamber. As Klaang addresses the High Council in Klingonese, Sato tries her best to interpret his words. According to the linguist, Klaang says something about disgracing the Klingon Empire and mentions that he's ready to die. The Klingon Chancellor approaches Klaang and makes a small cut in his hand with a jagged dagger. The Klingons then pour some of Klaang's blood into a vial which they examine with a large scientific device. The extracted DNA from his blood contains a wealth of Suliban information. The Klingons shout in gruff approval, but soon quiet again. The Chancellor approaches Captain Archer and, holding the dagger against the captain's throat, says something in Klingonese that Archer interprets as a thanks. Once the Chancellor leaves, Sato comments that the captain's interpretation was incorrect, and claims that Archer wouldn't want to know what the Klingon actually said.
It would seem that Klaang admitted he had disgraced the Empire. As we know well, the Klingon culture is based on honour. That Klaang was saved from death (what would have been an honourable death, working for the sake of the Empire) may have brought dishonour upon him.
Although we could interpret this scenario as being a generally reasonable first contact, for Klaang it didn't go down well. The Klingons probably weren't terribly impressed that they had to have Humans return their officer; it doesn't make the Empire look good.
Strike 2: An Unexpected Encounter
In ENT 'Unexpected', the Enterprise has a less than pleasant encounter with a group of Klingons. The encounter doesn't help to strengthen their relationships.
Strike 3: The Straw That Broke The Camel's... er Klingon's Back
In ENT 'Judgement', Archer is found guilty of assisting a group who were considered 'rebels' in the eyes of the Empire. He is then sentenced for life on Rura Penthe. Archer then breaks out of this and has a price put on his head.
Now, when we consider that Archer is essentially the Klingon's only experience with humans and that he is a representative of humans, I think it's fair to say that these actions didn't help the Human-Klingon relationship.
Whether it's fair to say this is down purely to the first contact is another matter. I would argue that the first contact wasn't the reason why Humans and Klingons went to war, but I do think it was at the least a contributing factor and, at most, a catalyst. This first contact in 'Broken Bow' did not portray the Empire as the strong force the Klingons would have wanted it to be.
It also depends on how you interpret it; I think @paxdiablo makes a good point; we could interpret Picard's statements as referring just to the first contact, or, as I have done, to the first contacts. I think the latter is more in the spirit of what Picard was trying to communicate.