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As a lengthy thread on Hacker News points out, this is probably the result of a legacy issue that dates back to the era of landline phones in the 1980s, and the number of letters that fit onto their dial pads.
Older landline telephones didn't have Q or Z on their keypads. So, using the letter Q or Z in a password would have been problematic for instances in which you needed to enter your account information over the phone.
There's no specific reason as to why older phones omitted those two letters, but some speculate that it could date back to the days in which telephone numbers became longer than five digits. There was concern that people would have a difficult time remembering phone numbers when they became longer than five digits, so each number from two to nine was assigned three letters. It's believed that Q and Z were left out because they look too similar to 0 and 2.
Look closely at this AT&T push button phone from the 1980s. You'll notice there's no Q or Z on the keypad.
Although Jet Blue's password FAQ page still states that you can't use these letters, the website seems to have updated its technology since. I created a Jet Blue True Blue account to see if this was indeed the case, and I had no issue creating a password with the letters Q and Z. Even so, the fact that such a dated requirement is still listed on Jet Blue's website indicates that it must have been in effect somewhat recently.
Since 2010, Jet Blue has been using the SABRE platform for reserving flights. SABRE is a software system for reserving hotels and flights that has been around since the 1950s, and it's hailed as being one of the world's first real-time operating systems.
Some have speculated that previous versions of SABRE didn't support the letters Q or Z in passwords since old-fashioned telephones didn't include those letters on their keypads. In its list of the worst password requirements from 2012, technology blog Kottke.org notes that the same strange rule was found in a website for one of SABRE's technology platforms, Sabre Red:
At the same time, however, a contributor in Stack Exchange's IT forums notes that Jet Blue used a different system called Open Skies before switching to SABRE, which could have been the source of the bizarre requirement.