If a pokemon uses protect, then detect, will the chance of failing still increase?
Yes. Protect, Detect, and Endure all have a chance of failure if Protect, Detect, or Endure was used on the previous turn.
Technically, the chance of failure doesn't increase; it is always 50% if Protect, Detect, or Endure was used in the previous turn. (I assume you're reffering to Gen IV because the chance of failure does indeed increase in older Generations.)