1. An ultrasonic measurement apparatus, comprising: a measurement data acquisition unit that transmits an ultrasonic wave to a body and acquires measurement data, which is obtained by measuring a cross section of the body with the ultrasonic wave, based on a received signal; and a processor configured to operate as: a displacement detection unit that detects displacement of a tissue in the body based on the measurement data, wherein the displacement detection unit detects the displacement of the tissue in the body in synchronization with a periodic contraction and relaxation of a target blood vessel, the displacement of the tissue in the body being tissue around the target blood vessel and being detected using a B-mode image that is one type of measurement data; and a blood vessel position detection unit that detects a cross-sectional center position of a blood vessel in the body based on the displacement of the tissue detected by the displacement detection unit.