There are a few words and phrases that you might be able to use. I'll show them to you in order of intimacy. In Japanese, the word "close" when referring to the relationship between two people or a group of people, it doesn't use the same word "close" that refers to the distance between two objects or people. Therefore it can't be directly translated. Here are 3 phrases I think might be useful:
良い関係がある （いいかんけい） - to have a good relationship
"I have a good relationship with my mother."
仲良い/仲が良い （なかよい/なかがいい）- close, friendly, on good terms
"He and tom get along well, don't they."
親しい (したしい) -close, intimate
"He is so close with his father that he's always in the same place his father is."
These are all different ways you can express the closeness of a relationship. I hope these help. If you have more questions, let me know.