Up front: I don't like "quality starts" as a statistic very much, and I think the reasons why are obvious. Quality starts are to actual quality starts as a colander is to a bowl. With that said, quality starts aren't completely worthless - just because they're littered with flaws doesn't mean they don't correlate well with effectiveness - and it seems we've got one guy chasing a team record. I bet you could never guess who that guy is.
Felix Hernandez has made 30 starts this season, and outside of a couple in May and one in early June, they have all met at least the minimum standards of quality. That's 27 quality starts out of 30 games, for a QS% of 90%. How does that rank in history? I went to Baseball-Reference, searched by team, added a minimum of 15 starts, and sorted Quality Start Percentage in descending order. 157 pitcher-seasons came out, and here's the tippy top:
1. Felix Hernandez, 2010, 90.0%
2. Felix Hernandez, 2009, 85.3%
3. Randy Johnson, 1997, 79.3%
4. Randy Johnson, 1994, 78.3%
5. Randy Johnson, 1995, 76.7%
In sixth is Jarrod Washburn's 2009, which is absurd.
So, Felix leads the way at 90%, narrowly ahead of himself at 85.3%. He's set to make another five starts on the season: @Anaheim, vs. Texas, @Toronto, @Texas, and vs. Oakland. Assuming he makes all five, the best he can do is 32/35, or 91.4%. If he wants to set a new record, he needs to go 3-for-5 the rest of the way, to finish at 30/35, or 85.7%. The worst he can do is 77.1%, which would still be good for fourth all-time.
It's also worth noting that, since 1961, among pitchers with at least 15 starts in a season, Felix's current quality start rate of 90% ranks seventh, league-wide, between Rick Reuschel's 1985 (92.3%) and Don Robinson's 1988 (89.5%). The best mark in modern baseball history belongs to Greg Maddux's 1994, at 96.0%. The worst? Bill Parsons' 1973, at 5.9%. That's one quality start out of 17, which kind of sounds even more pathetic than zero. Way to go, Bill.