It is unknown at this time. Jesper Christensen, the actor who portrayed Mr. White, described the Bond films he starred in as "****", and stated that "two movies are enough for me." Unless he changes his mind if asked to reprise the role, he will not return or he will be recast by another actor. While Daniel Craig denied that the next Bond movie(s) will continue the Quantum storyline back in 2008, he did state, "There is an organization that we can use whenever we want to."
The new screenwriter, John Logan, stated the upcoming Bond films he will write will build on what was established by "Skyfall". Since "Skyfall" had little to do with Mr. White or Quantum, it's unknown at this time if he wants to continue that storyline or if he will leave Mr. White's demise unresolved.