Monitor wakeup on mouse movement is something that windows users get, whereas Mac users get the best and when monitor is sleeping or is in low power state, a keyboard key press is required to wakeup the monitor. Mouse Movements / Clicks etc can be simulated by 3rd party software applications or even by a broken wired or wireless mouse and unwanted monitor wakeup is something which windows users still struggle.
The Idle time after which Monitor / Display should go into low power mode can be set using Energy Saver option of System Preferences. Have a look at the above screenshot in which the Display is configured to go blank or in low power mode whenever this Mac is left idle or inactive.
In order to wakeup the Monitor on Mac, you would need to press a key on keyboard or do a mouse click and only mouse movements will not wakeup the monitor from sleep mode. On the contrary, you can download a software from MurGaa.com to keep your Mac Computer always active by simulating automatic Mouse Cursor Movements of 1 pixel.