No. "Laconic" refers to speech. "Concise" usually refers to written material, and means "expressing a lot of information, accurately, in a small number of words or pages."
Here is an example of a use of the word "laconic."
"President Calvin Coolidge was a laconic man, known by the nickname 'Silent Cal.' At a dinner party, the hostess said to him, 'I have a bet with my husband. He says that I can't get you to say three words. I bet that I can.' Coolidge said 'You lose.'"
Here is an example of a use of the word "concise." "'The Elements of Style,' by Strunk and White, is only seventy pages long. But, though concise, it is one of the best guides to English composition you could ever read."