He was his agent when he signed a contract extension in 2012.
http://www.theargus.co.uk/sport/albion/9688135.Bridcutt_expected_to_extend_Albion_contract/?ref=rss
I've just been playing around on Google. Liam Bridcutt's agent was Mark Cartwright. Gus Poyet's agent was Athole Still. Maybe one of them has changed recently?
It doesn't appear that they do have the same agent.
I remember reading on Nsc that Bridge & Poyet had the same agent until someone...