The phrase "to blink an eye" means to react or respond quickly without hesitation. It suggests that someone didn't take much time to think or consider before taking action. Full definition
Just as most lawyers don't blink an eye at helping «guilty» criminals go unpunished, most physicians are not bothered by death. (rbs2.com)
but will go out to the movies and drop 60 bucks without blinking an eye for the same old plot lines played by different actors and the movie is usually 90 min at most. (tomsguide.com)
If you're prone to crying, bring your tissues for this one and if you're a hardcore zombie buff who didn't even blink an eye at The Walking Dead season 7 premiere, we bet you might let a few cries of woe out as well. (decider.com)