Sometimes the graphics card gets stuck outputting through either one of the HDMI/Display Ports etc or the monitor might be a bit fussy on how it picks up a new signal.
I have had a similar problem to you but managed to sort it thankfully so I hope this helps.
Steps to follow -
Switch your PC off firstly.
Connect both the HDMI lead and Display Port leads from your graphics card to your monitor so that the monitor has two potential inputs simultaneously.
Boot the PC and now the monitor should pick up at least one of the signals from either the HDMI or Display Port.
In your case it is more likely to pick up the Display Port as you said that works OK.
Use the OSD (On Screen Display) on your monitor to select the input and switch it to whichever one you prefer - you said it wasn't displaying through the HDMI port on your original post so I am presuming this is your desired one.
If the screen just goes blank or displays 'no signal' then go to 'Setting the HDMI input' steps below - but switch the PC off again first. You might just have to do this via the PC's front power switch as you will have no display if this is the case.
If you do get a signal working through the HDMI port this way it means the graphics card and monitor should have switched to the HDMI port. If so switch the PC off and unplug the Display Port lead both ends and boot it up.
If it works consistently then you have sorted it and can put the hammer away now.
If not then read on.....
Setting the HDMI input -
If it doesn't work through the HDMI input from the OSD on the monitor still then switch it off, leaving both the HDMI and Display Port leads plugged in both ends, and start the PC up.
It should work out of the Display Port as normal but if you get 'no signal' still then use the monitor's OSD to reset it so that it picks up its default input as normal when it worked before.
If you are still getting 'no signal' after doing this then unplug the HDMI lead both from the graphics card and monitor, reset the monitor via its OSD and boot up with just the Display Port connected to go back to when it was alright.
With both the HDMI and Display Port connected go into Windows and open 'Settings' (the little cog icon on the Start menu) , System, Display.
At the top of this page you should see 'rearrange your displays' and a window with a rectangle '1' and '2' in it to represent dual display. Note which numbered rectangle is highlighted in the window - either rectangle 1 or 2.
This is assuming Windows has picked up the two outputs of HDMI and Display Port & that your graphics card is feeding both signals.
Scroll to the bottom of the same page to 'Multiple Displays' and there should be a pull down menu with a few options including 'Show Only on 1' and 'Show Only on 2'.
Change this option to correspond with the rectangle number that is highlighted in the previous step at the top of the page - either 1 or 2.
This hopefully will set your graphics card to permanently display out of just one port and not attempt to activate the other.
If you only see only 1 display at the top of the screen in Settings with '1' in it then I'm not sure to be honest as either the card either isn't outputting via the HDMI at all as the previous poster Yuka said or Windows has problem detecting both inputs simultaneously.
You can undo this step to activate either/both at anytime though if you need to change the display input at a later date.
Hope it helps as I know it's frustrating sometimes!
Andy