As a native English Speaker if you said this to me in conversation I would understand it to mean you have some amount of confidence but not a lot.
It is quite common to hear people speak like this but I suspect the use of this sentence is more common as the reply to a question rather than a opening sentence.
Alice: "Are you confident with his answer?"
Bob: "I am not unconfident with his answer."
In my mind the sentence is quantifying the level of confidence you have. This list shows where I understand your confidence level to be."
"I am very confident"
"I am confident"
"I am somewhat confident"
"I am not unconfident"
"I am unconfident"