Is it permissible to ask a maternal cousin about the reason for her not greeting or for her feeling hatred, while observing the rules of forbidden seclusion (khalwa)?

The daughter of a maternal aunt is a non-mahram (a marriageable relative) and not one of the closer unmarriageable kin. It is permissible to speak with a non-mahram woman when there is a need, provided there is no fear of temptation (fitna). If temptation is feared, then it is not permissible to speak, even if it is merely to offer greetings of peace. The daughter of the maternal aunt might refrain from offering greetings of peace out of fear of temptation, not out of dislike. The questioner can speak to her father or some of her mahrams to ask them about this.