Hard to say at this point, and being intermittent doesn't make it any easier. I would say this could be the PSU, the monitor, the graphics solution or the motherboard. You need to eliminate the possibilities.
First, make sure the video cables are securely fastened - on both ends.
Any beeps? Without knowing the exact motherboard model number, I cannot tell if the board has an integrated piezoelectric "button" speaker but I don't see one on
this board. All cases used to have speakers. Then some motherboard makers started putting them on some (but not all) of their boards so case makers stopped including them with their cases.
That's too bad because beep codes can be a valuable troubleshooting aid. I don't know about the availability in Croatia but it is very inexpensive to add a
System Speaker to a computer. The ATX Form Factor standard does not require there be a speaker, but the standard does require motherboards have the header connectors to support one. They are located in the same front panel I/O header where the front panel power button connects. Note that price is for 20! I add one to all my builds is the motherboard does not have an integrated speaker.
The Intel H67 came out in 2011. It does not sound like a CMOS battery issue, but if it is more than 3 years old, I might swap it out since that is so easy to do. And they are inexpensive so little is lost if it does not fix the problem. Just remember to unplug the computer from the wall and touch bare metal of the case interior before reaching in to discharge any static in your body first. And don't touch the new battery with your bare fingers as skin oils promote corrosion and attract dust - I put a clean sock over my hand. Connect power and boot [if it will let you!] directly into the BIOS Setup menu to check/reset the date and time, and to make sure your boot order is where you want it.
Normally I point out that everything inside the computer depends on good, clean, stable power so I generally recommend swapping in a known good power supply to see if the problem occurs/remains. I still recommend that before spending money on anything else, but since this could also be a monitor problem and monitors are often easier to swap than PSUs, you might try a different monitor first. If no spare monitor, most TVs support computer input.
If you still get the same problem after this, I would look at your graphics solution. If you are using the on-board, you might try swapping interfaces (from HDMI to DVI for example). If same problem, then you will need to toss in a card. Hopefully you have a spare laying around. If still a problem, then it would appear to be a motherboard issue and it may be time to let this system go.