Extreme mass-ratio hyperbolic encounters (EMRHEs) around the supermassive black holes will be observable at the future gravitational-wave (GW) detectors in space, such as LISA and Taiji. Here we consider such EMRHEs in the presence of surrounding matter distribution including baryonic accretion disk and dark matter (DM) spike, and estimate their effects on the orbital evolution and GW waveforms. We find that large possible impacts come from the gravitational potential of the accretion disk, while the influence of the DM spike is small. We also illustrate that environments can leave distinctive imprints on the GW waveforms, but resolving such modifications is found to be challenging for LISA-like detectors in the near future.