Can an HDMI cable carry a mouse a keyboard signal?

No, HDMI cables don't support this functionality.

HDMI does have a backchannel (CEC) which in principle could be used to carry mouse and keyboard signals, but I don't know of any hardware which actually uses it for that. You may be able to use it that way in hardware you built yourself, but it's extremely unlikely your monitor supports it.

Monitors sometimes have an integrated USB hub for mouse and keyboard, and require an USB cable back to the computers.