None of the EQ effects currently present in FMOD Studio support sidechaining directly, but it is possible to obtain the behavior you describe by using a workaround.
The multiband EQ effect can be automated. This means you can add a sidechain modulator to the parameter used to automate the effect.
Support for more direct sidechaining of compressors is on our to-do list, but has not yet been scheduled for development. It is therefore unlikely to be released in time to be included in your project.