I often find myself in a similar situation, starting up a Q but cancelling it a minute in, because I no longer feel like playing.
I can find great joy in non-standard game modes though, such as ARAM or the rotating modes. It lets you play league without the every day problems. Its a nice change of pace and a bit more relaxing. Playing with premades works wonders too, I find a lot more joy in Q'ing up with friends than playing solo.
Changing the role you play may help you as well. I recently switched from toplane to support, and overall it has been a great experience. The play style is very different; and as I'm relatively new to and still bad at the role, I feel motivated to improve and try new champions.
That said there's nothing wrong with taking breaks. I have been playing since S2 but it hardly feels that way, because of the long breaks I took. If you're tired of LoL, just stop playing for a few months and come back refreshed. Even taking a few days off to play some other games can probably already do some good!