I doubt it. He's a lot of fun to watch and is steadily improving (his completion % was a bit ropey last year, but he's turned that around), but he's undersized for a starting Sunday QB. That said, he is tough as old boots. He got battered last season and saw the games out with a torn rotor cuff (which it looked like he might have suffered against the Cowboys, so hid and played with for a while). But in the pros he'd get swamped. He'd have the same problems RGIII has, and he doesn't have Griffin's ball skills.
He's in a pass-happy team, so he'll be on TV a bit this year, and him and Kevin White should put up some nice looking numbers. I reckon he'll have a similar path as Seth Doege last year - he'll decalre and probably go undrafted, signing up somewhere as a FA afterwards. I wouldn't expect him to make it past the preseason though. At a stretch he might make a practice roster, but I doubt it. There's not a lot of value in him for an NFL side. Doege plays in the CFL though, and if Trickett wants to play pro ball for a while, that would be a viable option.