Does Page verification require downtime?
Nope, no downtime for this change at all. 

Here's the explanation of what/when things happen:


  Keep in mind that just changing the setting to CHECKSUM does not
  instantly add CHECKSUMs to your existing data pages in the database.
  In order for this to happen, you have to read each page into memory,
  make some sort of change and then write it back out to the storage
  subsystem. This can happen from normal INSERT/UPDATE/DELETE activity
  over time, or from rebuilding your indexes.


Source: http://www.sqlskills.com/blogs/glenn/setting-your-page-verify-database-option-to-checksum/