This is in Chapter 10, the same chapter where Atticus (and Miss Maudie) tell Scout that it is a sin to kill a mockingbird. Scout mentions that this is the only time Atticus says it was a sin to do something. So, Atticus only shoots Tim Johnson because he has no other choice. Atticus didn't want to kill the dog; but he had to. So, he certainly wouldn't brag about something he didn't want to do. Atticus is too humble to brag about his skills. Even as a lawyer and a father he never brags; he just lives his life, let all others think what they may.
More to the point, Atticus is not the kind of man who would want to brag about a violent skill; especially to his children for whom he would want to set a good example. As Jem starts to mature, he recognizes this and finishes the chapter by saying, "Atticus is a gentleman, just like me!" Jem no longer cares if Atticus is older than the other boys' fathers. This is significant as a sign of Jem's maturity. Scout wants to brag about her father. But Jem learns another of this novel's lessons which is that if you do the right thing, it doesn't matter what others think. Jem recognizes that it is unnecessary and pretentious to brag.
Jem does not want Scout to boast about Atticus's having shot the rabid dog because he feels that their father would not approve of such behavior.
After Zeebo throws the dead dog into the garbage truck, the children head home. Scout tells Jem excitedly that they will have quite a tale to tell at school on Monday, boasting, "Ain't everybody's daddy the deadest shot in Maycomb County." But Jem urges her not to say anything about the incident. Further, he tells Scout that if Atticus had wanted them to know that he was a great shot, he would have told them. And, if he were proud of his shot, he would have told them also. Therefore, it is obvious that Atticus likes to keep his shooting talent hidden.
Without understanding, Scout suggests that perhaps Atticus has just forgotten to mention his being an expert shot. With brotherly superiority, Jem informs her that she would not understand. Picking up a stone, Jem throws it with triumphant satisfaction. Then, in a display of his new appreciation for his father, he runs and calls back to Scout, "Atticus is a gentleman, just like me!"