I have a close relationship with my mom. 　私は母と良い関係です。/Watashi wa haha to yoi/ii kankei desu. My mom and I are very close. 　母とはとても仲が良いです。/Haha towa totemo naka ga yoi/ii desu. You can also say '母と私はとても.....'. My siblings and I are close in age. 　兄弟と年が近いです。/ Kyoudai to toshi ga chikai desu. Close relationship as in; family　仲が良い、良い関係 friend, boyfriend or girlfriend　仲が良い、仲良し、良い関係、親しい(formal expression)、親しくさせていただいている(very formal expression)

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 Ex: お母さんと良い関係があります。 "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 Ex: 彼とトムは仲が良いですね。 "He and tom get along well, don't they." 親しい (したしい) -close, intimate Ex: 彼はいつもお父さんのところにいるほど親しいです。 "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.

ありがとうございます。

close relationship　「仲が良い」「とても仲が良い」じゃダメ？　casualなら「ちょー仲よし」 あとは　「～のように仲が良い」　、双子、兄弟、姉妹 close in age.　「歳が近い」「同年代」