The suggested Yamaha RX-A1010 receiver seems very nice, but, in practice, the power difference between that one and the Denon AVR-1912 is relatively inconsequential:
Denon AVR-1912, 90 watts per channel (8 Ohm - 20 - 20000 Hz - THD 0.08 %)
Yamaha RX-A1010, 110 watts per channel (8 Ohm - 20 - 20000 Hz - THD 0.06 %)
Basically, the Yamaha makes about 22% more power (and 30% more dynamic power). That's nice, but it's not related to your current problem.
(A 30% difference in dynamic power might be just enough, perhaps, to notice that one system is able to play slightly louder than the other. It's minor.)
The problem you're having is more likely to be related to the small, inefficient speakers you're using.
The Q acoustics 2000i Cinema Pack speakers seems pretty cool, but the small size of the satellite speakers suggest they would work better in a smaller sized room, like a bedroom, played at moderate volume levels. They are not suitable if you're trying to use them in a larger size room, and want them to play at higher volume levels.
The 2010i bookshelf speakers (which you have four of), have small 4" (100mm) woofers, and the sensitivity rating is 86 dB, which is low. (Low sensitivity means they won't play as loudly --or play loudly as easily --as speakers that are more efficient.)
The maximum recommended power for the 2010i speakers is 75 watts per channel. This is a problem, not just because the Denon receiver is rated at 90 watts per channel (into 8 Ohms), but also because the 2010i speakers are rated at 6 Ohms (with a minimum of 4 Ohms). That will cause them to draw even more power from the Denon receiver (as much as 125 watts).
So, if you've been trying to make the system play even somewhat loudly, in a good sized room, the Denon receiver is probably overwhelming the small Q acoustics speakers. The distortion noise you were hearing could be the mechanical noise of the speakers bottoming out (or port noise that becomes audible).
You're fighting physics when you try to make very small speakers play loudly. They're intrinsically less efficient. Plus, small speakers often demonstrate a "compression effect," where, above a certain point, you apply more and more power, but they just don't play any louder. (This can also change the frequency response.)
Basically, it just seems like the Q acoustics 2000i speakers aren't quite big enough for the way you've been using them.
So, if that seems to be the situation --if you have a pretty big room, and you need your home theater system to play at fairly high levels --you should move the Q acoustics speakers to a bedroom system, and then plan on buying some equally good quality, but physically somewhat larger, speakers, to use with your main system. Speakers that are physically larger, and more efficient, will be able to fill the space with sound more effortlessly --louder, with better dynamics --and will be a better match (in terms of power handling) for your current Denon receiver.
Edit,
Do you have the port plugs installed? If you keep the port plugs installed in the Q acoustics 2010i speakers it will probably allow them to absorb a little more power before producing audible distress.
Otherwise, until you upgrade, if you notice distortion sounds coming from the speakers, and the problem is immediately solved by turning down the volume control a little bit, that's all you need to do.