What cable do I need for 2560 x 1440 resolution on my Mac Air mid 2013? My LG Monitor has HDMI input only
You need an HDMI cable that conforms to the HDMI 1.3 (minimum) specification.  When looking for this adapter, be sure to specify "4K" so that it brings up relevant results.  The Plugable Active Mini DisplayPort to HDMI 2.0 Adapter will work for your application.

That doesn't guarantee that you will display 2560x1440.  Your monitor must support it as well.

Reviewing the specs of the MacBook Air, it will support resolutions on an external monitor of up to 2560x1600 so you will have no problem here.