We present results of computational quadratic Chabauty experiments on genus 2 bielliptic modular curves of Jacobian rank 1 and 2 that have recently been added to the LMFDB. We work over both the rationals and quadratic imaginary fields. In several cases, the experiments yielded algebraic irrational points among the set of mock rational points. We highlight several notable examples and offer a conjecture relating the level of the modular curve to the potential number fields over which these points can arise.