Disclosed is an apparatus for measuring convergence using a fiber Bragg grating sensor measuring slight deformation, capable of measuring the convergence such as the inclination or the gravitational acceleration of a measurement target including a bridge or a tunnel by measuring the deformation of the fiber Bragg grating sensor when the measurement target is inclined or vibrated, as well as a measurement method using the apparatus. The apparatus includes a main body that is installed on a measurement target; a fixture that is fixedly installed at the front of the main body; a rotator that is horizontally installed apart from the fixture at a predetermined distance and is rotatably installed in the main body using a bearing; an optical fiber that is adhered to upper outer surfaces of the fixture and the rotator so as to maintain a horizontally tensioned state and has optical terminals installed on opposite ends thereof; a fiber Bragg grating sensor that is installed on the optical fiber between the fixture and the rotator; and a bob that is vertically connected with a lower portion of the rotator by a support having a predetermined length. The apparatus is installed on the measurement target after setting a reference value and then calculates a quantity of displacement caused by variation of the ground, so that values of angular displacement and gravitational acceleration displacement are measured.