It depends.  Using the Air Force as an example, in official correspondence — that is, email intended to conduct official military business — it is not technically permissible for members to have quotes of any kind in their signature blocks.  This is “technically” true because this rule, while in place, is rarely enforced.

Air Force Instruction (AFI) 33-119 says that 

Users will not add slogans, quotes, special backgrounds, special stationeries, digital images, unusual fonts, etc., to the body of their electronic messages.

Thus, a strict application of this AFI means that no one is allowed to have any quote of any content in their official email signature block.  Still, it is not all that unusual to see quotes from Vince Lombardi, Neo (The Matrix), Patton, Buddha, Jesus, or any of a host others in email signature blocks from military addresses.

This rule refers specifically to official email (which is the context of the AFI). While possible, it is unlikely that this rule would be (properly) applied against a military member who used their email for an authorized unofficial use.  Some civilians (and some military personnel) may be surprised to learn a large percentage of military email is actually “unofficial.”

At present, the Navy and Army do not appear to have explicit or readily available policies on quotations in email signature blocks.  In all cases, local policies and commander directives can always be more restrictive.

Update: This discussion is about what one is allowed to do; whether or not one should do something (that is otherwise permissible) is always a matter of prayer, good judgment, and military professionalism.

This question is part of the ChristianFighterPilot.com FAQ on religion and the military.

See a discussion on the topic of religious content and military email.