Since Brooklyn Nine-Nine is a show about an NYPD precinct and all, you probably should find a legal way to watch the first season. It just wouldn't feel right otherwise. So do your civic duty and try one of these methods to catch up before Season 2 starts.
FOX On Demand or Online
FOX still has its Fall Binge promotion going on, where you can catch up on your favorite shows from the network before their respective seasons begin this month and next. However, with only five episodes currently available to view On Demand through most cable providers and on FOX.com, it isn't so much a binge as a meal where you ate a little bit more than usual.
Other Online Streaming Services
If you truly want to watch the whole season of Brooklyn Nine-Nine, you've got to pay for it. You can buy the full seasons, or individual episodes if you only missed a select few, on iTunes and Amazon Instant Video. However, Hulu might be your best bet. While you can only get those same five episodes that are available On Demand and on FOX.com for free, you can catch the whole series by connecting a participating cable provider to your account or with a Hulu Plus subscription. In case you don't already shell out the $7.99 per month, you can sign up right now and get one week for free. Since there are only 22 episodes, and each one is only about 22 minutes long, you can certainly power through the whole season in a weekend. They don't call it binge-watching for nothing.
Just Start With Season 2
So it would be a bummer to miss out on the stellar first season of this show. It truly would. But Brooklyn Nine-Nine isn't Breaking Bad or anything like that, so if you just started with the Season 2 premiere, I'm sure you'd be able to catch on to what has happened on the show pretty quickly. And if there's anything that doesn't make sense, use your sleuthing skills and just Google it.