Use an external display with your MacBook Air - Apple Support
"MacBook Air with M3 chip can support up to two external displays when the laptop lid is closed. The primary display can support up to 6K resolution at 60 Hz or 4K resolution at 144 Hz, and the secondary display can support up to 5K resolution at 60 Hz or 4K resolution at 100 Hz."
This article indicates that for a monitor with 4K resolution, you can get higher refresh rates (144 Hz or 100 Hz) than the Technical Specifications promise. There is no indication even here that a plain M3 chip supports 240 Hz refresh rates for any monitor.
My guess is that you'll be able to connect the monitor at 60 Hz – maybe 120 Hz, if you're lucky. But not 240 Hz. If the manufacturer of your monitor has manuals on their support site, the manual might contain "preset mode" tables showing you which combinations of inputs, resolutions, and refresh rates the monitor supports.