My iMac won't enter "Target Display Mode" for my MacBook Air

We need information about both machines. Apple added restrictions on the age of the new machine.

if the other machine is a “MacBook Air 13″, macOS 12.6” as suggested by your tag line, then its version of macOS is “too new” .., it does not support connecting to another Mac that is running in Target Display Mode.

There would also be a problem if the other machine had a hardware model year later than 2019, or if the connection was anything but a Thunderbolt connection between Thunderbolt ports. The Thunderbolt requirement is a technical one and always existed (for your 2011 iMac), while the other restrictions came along later.

What model iMac and what version of MacOS is it running?

Use your iMac as a display with target display mode - Apple Support

If it is a 2011 iMac running High Sierra, it should be able to run in Thunderbolt Target Display Mode.

What computer are you trying to use it with? What is the hardware model and model year of that machine? Is the connection from one Thunderbolt port to another, using a Thunderbolt cable?

”m trying to connect from thunderbolt to hdmi adapter to have the Mac as a monitor for my Xbox.”

That definitely won’t work. You can’t convert HDMI to Thunderbolt. I’m guessing that you are trying to use a Mini DisplayPort to HDMI adapter backwards. Thunderbolt 1 & 2 used the Mini DisplayPort connector, and Thunderbolt 1 & 2 host points on Macs could pretend to be Minu DisplayPorts for the purpose of driving external monitors. But Thunderbolt was designed as a more general purpose expansion interface than DisplayPort or HDMI.

If you need a dedicated screen for the Xbox, your best route might be to get a 1080p (1920x1080 pixel) or Ultra HD 4K (3840x2160 pixel) TV set or monitor. I don't know, offhand, if you have one of the Xbox models that support 4K screens; I'm mentioning the possibility in case you have an Xbox that does.

https://support.xbox.com/en-US/help/hardware-network/display-sound/do-my-xbox-and-tv-support-4k

If you need something that will work both with your MacBook Air and with your Xbox, you might want to look at computer monitors that have a resolution that your Xbox supports, and that have multiple inputs – at least one being a HDMI input.

For your Xbox, you are going to need a real TV set or standalone computer monitor.