It’s not that God caused David to do wrong. David was capable of choosing right or wrong. God permitted David to pursue his sinful choices and reap the consequences of his actions. To incite in this case may have meant that God orchestrated the events that led to David’s decision.
To make the question even more confusing, the writer of 1 Chronicles (21:1) says it was not God but Satan that incited David to do wrong. The apparent conflict between the two passages may mean that God sometimes permits Satan to do things that subsequently are viewed as having come from God.
The writer of 2 Samuel reports that God was ultimately behind David’s action, highlighting God’s sovereignty. The writer of 1 Chronicles recognizes Satan as the more immediate temptation behind David’s decision. Both views are correct.
In the end, God used David’s sinful desires to bring judgment for sin upon the nation of Israel. But good also came from these sad events: Because of what happened here, the site was acquired where the temple eventually would be built.