The Mini DisplayPort/Thunderbolt 2 port on the 2017 Air is "pickier" about adapters and HDMI handshakes than the earlier models. The fact that the TV reports “no video signal” usually means the Mac isn’t successfully negotiating the display mode with the TV.
First, the most likely cause is the adapter itself. Many Mini DisplayPort → HDMI adapters are passive and rely on older signaling that worked fine on earlier Air models. The 2017 often requires an active Mini DisplayPort to HDMI adapter, especially with TVs. If the adapter is unbranded or older, try a different one, preferably one that explicitly states compatibility with macOS Sierra or later.
Next, try forcing the display output manually. Connect everything, then go to System Preferences → Displays and hold the Option key while clicking Detect Displays. (Sorry, I don't have a Mac running Monterey available to confirm the exact menu items.) If the TV appears, set the resolution manually to 1920×1080 at 60 Hz. Toshiba Regza TVs are known to reject non-standard refresh rates or higher resolutions during HDMI negotiation, which can result in a blank screen even though the cable is connected.
If that still doesn’t work, shut down the MacBook completely, connect the HDMI cable to the TV first, then power on the TV, and finally boot the MacBook Air. This forces the Mac to detect the external display during startup. Also try resetting the SMC and NVRAM, as display output settings are stored there and can occasionally break when moving between machines.