Does the Battlestar Galactica game spoil the TV series?
Absolutely

(Very) broadly speaking, the base game spoils Season 1, Pegasus spoils Season 2 (and the start of Season 3) and Exodus spoils Seasons 3 and 4.

At least one exception to this that I can remember is that one Pegasus character's card spoils an event in Season 4. (Spoiler follows)


   Dualla's disadvantage


Having said that, I've played with multiple people who were watching the series at the same time. As long as you're roughly half-way through the first season, you should be (mostly) fine for the base set (as long as you don't read too much flavour text!). I'd hold off on the expansions until you've seen a lot more though.