The major external conflict in the story is the ongoing struggle between the Musketeers and Cardinal Richelieu's guards. This external conflict is related to D'Artagnan's own inner conflict. He's young, impetuous, and filled with pride. Shortly after he arrives in Paris his pride gets the better of him as he challenges, in turn, each of the Three Musketeers to a duel. Thankfully, for D'Artagnan, the arrival of the cardinal's guards gives him a chance to channel his pride in a more positive direction, joining with the Musketeers to fight against them. To some extent, D'Artagnan's inner conflict between his youth and his desire to become a man of honor is resolved.
Another conflict is still troubling D'Artagnan, though. He's just a humble country boy from a poor background and so naturally feels a sense of social inferiority to the aristocrats whose world he now inhabits. D'Artagnan is naturally brave, if still occasionally foolhardy, but there's a sense that he's constantly trying to prove himself—to show the world that, whatever his background, he has nobility of spirit.
In all his actions, he is playing the role of aristocrat. Whether it's saving Constance from being kidnapped, or retrieving the Queen's jewels to save her from public disgrace, or even just engaging in a brief skirmish with the cardinal's men, D'Artagnan is actively playing up to the role of noble blade. And it's not until the end of the swashbuckling tale, when D'Artagnan is finally commissioned as a lieutenant in the Musketeers, that the conflict between the naive country boy and his aristocratic self-image is fully resolved.