This is dependent on the specific monitor, not the HDMI version, as all versions of HDMI currently in use are capable of 1080p 120 Hz, the last HDMI version that wasn't capable of this was HDMI 1.2, which was deprecated in 2006 and hasn't been used in new devices for many years. HDMI version 1.3 and above allow at least up to 1080p 144 Hz (240 Hz for HDMI 2.0).
However, HDMI devices are not required to implement the maximum capability of whatever HDMI version they are using, so monitors may have their own limitations which are lower than the limitations of HDMI itself. Some monitors with HDMI 1.4 can do 1080p 144 Hz, most monitors with HDMI 1.4 can't. Some monitors with HDMI 2.0 will be able to reach 144 Hz, some monitors with HDMI 2.0 may only allow up to 60 Hz.
The Acer GN246HL has an HDMI 1.4a port, but does not allow higher than 60 Hz over HDMI.