I lived in Avon Lake for over 4 years and went to CQ many times. Harry, the owner and his son are great bartenders and Tommy Murphy, another bartender was my next door neighbor's brother.
First "discovered" CQ on Christmas Eve, a number of years ago. That is customer appreciation day and the place is always elbow to elbow. It truly is "close quarters" since the bar is only about 8 feet wide and 20 feet long. On Christmas Eve the first beer is free and they hand you a mini bottle of Irish Creme.
Guinness and Smithwick's on tap.
Limited menu but everything is freshly prepared - you can watch Harry make your fries from scratch (get the spicy fries - they are excellent). The wings are totally awesome - get them "dirty" and their pulled pork sandwich is not bad either.
The olives on the bar are incredible!!
You cannot go in there without expecting to meet someone you don't know and having a great conversation. We were there one night and a woman walked in with cookies her mother had baked - shared them with the bar.
Another thing - Harry will take your picture and post it on the bulletin board. My wife and I have a couple of pictures from our visits since Harry wants you to take them.
Voted "Best Dive Bar" in cleveland magazine in 2005.
Do yourself a favor and stop by some time when you are in Cleveland. Not open on Sunday and don't be surprised to see the place closed on the spur of the moment if Harry decides to take off for a week..