"Confide" works well only if you are speaking particularly of private information that one wouldn't tell nearly anyone. We nearly always "confide (something) in" our best friends, not "confide to": "I know that I can confide in you".
I think I nearly always resort to descriptions for this; I might write "We had a deep, personal conversation" or "I wish I had someone that I could really talk to about things dear to me.
One common adjective is "heart-to-heart", for example: "I tried to have a heart-to-heart conversation with my wife, but she was too tired to take me seriously." This usually connotes that there is a certain topic that needs to be discussed, unless it is used with more generalizing context clues, such as "I really like talking with Ann; our chats are always so heart-to-heart."