According to TechPowerUp on the Radeon RX 580 it has the following video ports:
Display outputs include: 1x HDMI 2.0b, 3x DisplayPort 1.4a.
Google DisplayPort version 1.4a to get the maximum Resolutions and Frequencies that is supported by that version.
Then Google HDMI version 2.0b to the the maximum Resolutions and Frequencies that is supported by that version.
Then you will know if your GPU card supports 240Hz at which frequencies and dual displays at which resolutions and frequencies.