My late father was a member of the Quiet Birdmen and I would like to
get in touch with their national chapter. Do you know how I can do
thins?

The Quiet Birdmen are very secretive and reclusive. Local chapters
(called "hangars") that have websites typically require passwords for
access, and I have found no evidence that there is a
publicly-accessible national site.

One of the few local chapters that still has a public site is this
one, but its home page says that it will soon be password-protected:

http://www.gwwquietbirdmen.com/index.htm

I found this posted on a messageboard, but I cannot vouch for its accuracy:

"For those who asked, the Quiet Birdmen are still active.

 If you wish to contact them try:

 1 Earhart Drive
 Suite 5
 Coatesville, PA 
 19320

(610) 380-4981"

http://www.charleslindbergh.com/cgi-bin/ultimatebb.cgi?ubb=reply;f=1;t=000590

The site linked in pinkfreud-ga's comment links to the
password-protected national site, qbird.org.  A WHOIS on the domain
qbird.org gives the following registrant information:

Quiet Birdmen, Inc.
hq@qbird.org
1 Earhart Dr. Suite 5
Coatesville, PA 19320-2700
US
Phone: 610-380-4981

In other words it's pretty certain that the address and phone number
in the first comment are correct.