Nonlocal correlations observed from entangled quantum particles imply the existence of intrinsic randomness. Normally, locally projective measurements performed on a two-qubit entangled state can only certify one-bit randomness at most, while non-projective measurement can certify more randomness with the same quantum resources. In this Letter, we carry out an experimental investigation on quantum randomness certification through a symmetric informationally complete positive operator-valued measurement, which in principle can certify the maximum randomness through an entangled qubit. We observe the quantum nonlocal correlations that are close to the theoretical values. In the future, this work can provide a valuable reference for the research on the limit of randomness certification.