Look at your motherboard manual since you're using integrated graphics. Determine first if you can in fact use all 3 ports to run three monitors or if two of the ports are tied together so that you can only use two monitors. For example, sometimes the VGA + DVI ports can use only one of them, so you could do HDMI + DVI or HDMI + VGA but not all three. Others may have DVI and HDMI tied together, since they're both digital, so you could only do DVI + VGA or HDMI + VGA. Also check that the CPU you have allows you to use all three.
If it will run all 3 outputs together, then you should be able to use a DVI -> HDMI cable as the electrical signal is compatible it's just the physical connector that's different. Of course HDMI also supports audio whereas DVI doesn't so you can convert DVI to HDMI but when you convert HDMI to DVI you lose any audio that was sent over HDMI.
Of course if one of your existing monitor has DVI input, you could use the DVI output for that monitor and use the HDMI output for the third monitor so there's no converter cables.