There are serveral reasons why the narrator keeps her condition a secret from John and his sister.
First, realize that back in the setting of the story there was nothing known to the medical nor psychological field such as "Post Partum Depression" or "Baby Blues", which is a very true condition based on hormonal shifts, stimulus to the frontal cortex, depletion of minerals or excess of secreted hormones to the body. The result of this body imbalance is a deep-rooted depression combined with separation anxiety, anxiety attacks, tremors, irregular heart beats and difficulty breathing. Ultimately, it could also lead to hallucinations, unreasonable fears, and even suicide.
Back in the time where the story is set, women's conditions were reduced to "melancholy" "sadness" "joy that kills" and things that just solidified the idea that women were nothing but a bunch of nerves who made babies.
The result of her telling them would have been spending her days alone in a sanatorium, becoming isolated from her family, and bringing shame to those around her. She, herself, did not know either what was happening. She must have been a wreck knowing how irrational her thoughts were becoming, but she must have been even more scared to be declared "crazy".
As you also know, John was not the most affectionate nor devoted husband. Taking her away to take airs and relax in the country was just a way for him to rid himself of the problem by doing something that looks merciful. He was not at all interested in understanding what was wrong, nor did anything out of his way to try to figure it out.
Hence, the lack of support, the lack of understanding of her situation, and the horror and anxiety that awaits giving up to it were just some of the reasons why it was best to keep the whole ordeal a secret.