There are a lot of characters, events and subplots in the first and second books that aren't in the TV series, and the course of events is somewhat different, so while the broad strokes are the same, reading the second without having read the first will be a more confusing experience than otherwise.
The first book to first series is a close adaptation. There might be a few things and characters that don't make sense, but on the whole you should be fine starting in book 2.
My wife did it and she had maybe 2 questions starting from book 2 after watching the series
I urge you to read from book 1. The first season of the show was very accurate, but there's a lot missing. Bran's dreams are much more important and explicit. Ned's memories from the war. I think at least half of the lines attributed to most characters are from their own thoughts and the show never shows their thoughts with some echoey voice (thank R'hllor, it would be awful), so you miss a lot. You'd miss most of the background. There are also many characters that haven't appeared on the show that were introduced in the first book and return on the second one. I strongly recommend you to read from the very start.
I think that's what I will do. I couldn't resist the temptation of finishing the second season, but right now I am starting the first book. I will have to read an electronic version on a library's ipad or kindle, even though I prefer hard copies.
When I started reading, I couldn't help trying to remember what I watched in the series, but I managed to stop myself from doing that so that I can re-imagine everything myself.
I think it would be interesting to make a post about techniques to put aside "spoilers" you have already been exposed to when reading the books for the first time.