It would appear the answer is no; Bird has previously said in an interview he's never read Watchmen and has only heard of it in broad terms so it's unlikely it influenced his "No capes!" idea.
Barrier: I've been astonished by how precise the parallels have been that some people have drawn between the film and certain superhero comic books, like Powers, Watchmen, and Fantastic Four. I gather from other interviews, though, that you really haven't been that much of a comic-book reader, and really haven't been consciously influenced by these comic books. What kind of feedback have you been getting from fans about these supposed influences, and how have you been responding?
Bird: I was not a big comic-book reader. I read a few, when I was little, but I was really much more into things like "Peanuts" and "B.C."—funny strips. I got my heroes secondhand, from television and movies, to a certain extent. When fans ask if I was influenced by issue 47 of Whoeverman, I have no idea what they're talking about. I'm perfectly willing to believe that I'm not the first to come up with certain ideas involving superheroes; it's probably the most well-trod turf on the planet. If there are similarities, it's simply because the same thoughts that occurred to other people also occurred to me. I'd be astonished if anyone could come up with any truly original powers that were at all interesting any more.
That's not the part of the story that I'm interested in, anyway. The part that I'm interested in is all the personal stuff. I tried to base the powers on family archetypes. The father is always expected to be strong, so I had him have strength. Moms are always pulled in a million different directions, so I had her be elastic. Teenagers are insecure and defensive, so I had her be invisible and have protective shields. Ten-year-old boys are hyperactive energy balls, so I had him be speed. And babies are unknown—they may have great powers, they may have none.
Barrier: Had you even heard of Powers before?
Bird: No, no. I've heard of Watchmen. Other people have mentioned that aspects of it are similar to Incredibles, I think something about the superheroes being retired. I know it's very highly regarded; if you're going to be compared to something, it's nice if it's something good.
Michael Barrier.com, INTERVIEWS - Brad Bird
While I'm personally leaning towards it being Bird's own idea, the more "official" answer on the matter would appear to be just: we don't know. What I can say is the parallel between the two has been made before, however, neither add evidence that this was in fact the case.
Remember that iconic moment where Edna Mode rants about the dangers of wearing capes? Well, she is not wrong.
In a brief moment, WATCHMEN depicts a sequence that recounts the deaths of numerous vigilantes. Among these deaths includes that of Dollar Bill. While running after thieves, his cape gets stuck in a revolving door. The thieves then shoot and kill him at point blank range. It is quite the anticlimactic demise for a renowned vigilante, but he never thought about the risks of wearing a cape with his supersuit. Perhaps if Edna Mode existed in the WATCHMEN universe, he could have lived another day to fight crime. Poor guy.
Comicsverse, Real News: THE INCREDIBLES and WATCHMEN Are The Same Movie
Edna's explanation of capes getting superheroes in trouble is a nod to the graphic novel Watchmen wherein the character Dollar Bill is killed by bank robbers after his cape becomes stuck in a revolving door.
IMDb, The Incredibles, Trivia
The theory is possible that Bird was inspired by Dollar Bill considering that Dollar Bill's death initially showed up in Watchmen Vol 1 #2 from 1986 and Bird apparently had the idea for The Incredibles in 1993.
Edna as a character herself is a bit mysterious, apart from himself, Bird has been quite cagey when asked who she was inspired by. There are leading theories that she is inspired by Edith Head and Rei Kawakubo, among others. At least the Edith Head theory has been shot down, though he doesn't say she isn't a partial inspiration.
@BradBirdA113: “@oziecargile: Is Edna Mode really Edith Head?”
No. People from MANY countries (27 & counting) were convinced I based E on a local icon.
Twitter, @BradBirdA113 tweet
Further from that the most we really know about the eccentric designer is that she is half German and half Japanese. In the below interview Bird also lends credence to the theory that "No capes!" was his idea seeing as the main ideas behind the character are his.
How did you come up with this interesting, quirky character?
She was my most fun character to write. Any day that I was writing her, I was one happy camper. The idea was that superheroes always have these flamboyant costumes, and nobody explains who designs them. Every once in a while there would be a half-hearted attempt where they'd show some muscle-bound guy sewing in the basement. And I never really bought it that suddenly this guy had an interest in fashion, you know? So I thought if you had a world populated with superheroes like we do, that somebody would be designing this stuff. And she couldn't just be a designer, she would also have to be half scientist, sort of the technical whiz that outfits the hero in an action movie. "If you press this, it's a lighter, but it's also a rocket!"
What was the inspiration for that outrageously funny voice?
The reason the accent was sort of half Japanese and half German was they're two small countries that have amazing design and amazing technology. You think about the best cameras or cars or anything, they're German and Japanese. So that was just the goal. And they're small countries that have a big impact, so she's a tiny character that dominates the room when she gets into it. We tried to make her house huge and she's tiny, but she fills it, you know? I remember years and years and years ago, I met Bette Midler, and I was shocked at how small she was. Because when she's onscreen, she absolutely dominates the screen. And it just struck me how much personality was in this small body.
RadioFree.com, Interviews: Brad Bird, The Incredibles
With the character being this mysterious it is hard to really comment on one particular aspect of her as we know little of her generally. However, I have reached out to Brad for an answer so will have to see if I ever get a reply.