Is it forbidden to speak with a female relative with the intention of avoiding family problems, with her mother's knowledge, and without delving into matters that provoke desire?

If you are able to marry this relative, then proceed with it. Until marriage becomes feasible, adhere to Allah's boundaries and maintain the manners of Islamic law in dealing with non-mahram (definition) women. You are not permitted to speak to her or correspond with her without a valid need, whether with or without her family's knowledge. This is a door to temptation and a key to evil. Al-Khadimi said: "Speaking with a young non-mahram woman is not permissible without need, as it is a cause for temptation." And beware of being complacent in this matter by claiming that you are not speaking about anything that incites lust, or that you intend by speaking with her to avoid family problems. All of this could be from the steps of Satan.