Just happened to me in an ARAM game. For some reason, my Q and R stopped working completely. I couldn't click or use the hotkeys. I tried resetting key bindings, assigning new keys, etc. I also tried alt-tabbing out and tabbing back in...nothing seemed to work. Both Q and R hotkeys worked fine when I was in Sion's undead state, though.
I'm on Windows 7
I sent a message about this to Riot, and they replied back with lots of useful facts & details. The main cause of this bug to happened is due to
To fix this, set your settings to Quick Cast, and try testing your Q. If it works, set it back to Normal casts and play as usual. It works for me before. This was already a known problem to Riot, and they said it is a very hard & "stubborn" bug to fix.
I hade the same exact issue just happen in ARAM, I am playing on widows 8.1 and was broadcasting the game, twitch.tv is now exporting the video to youtube and Ill add link for entire game in a few mins, but here is the twitch link for now for visual verification of everything that went down.
http://www.twitch.tv/voggaming/b/626302996
As you can also hear, the game was hearing my commands to fire these abilities with clicks, but the abilities just would not work.
I've had this issue in the past though not recently so I figured they fixed it. To use q and r again simply exit the game and reconnect. It's definitely a pain but well worth the temporary dc to be able to use half your skills...
Here is youtube upload. http://youtu.be/GTczkJzRdKI The issues start at 25min and 30seconds (25:30) Annotations added in video for more info and to show whats going on.
A problem with champion Qs suddenly not working seems to be popping up everywhere; I've seen like 5 posts about champion Qs going haywire. It's been a different champ each time though. I have the same issue right now but specifically with Varus; it happens when I have maxed out his Q skill - is it happening to you at the same time?
Hey! I was playing Kat 
in a 5v5, and I had the same exact problem! Recall doesn't work either, but it doesn't affect you in ARAM so you wouldn't see. So this is a reoccurring problem!
