I had to do this just now and I thought I would document it (at least for myself) and maybe it will help others.

This probably isn't all the things everyone might have to do but it works for how I use CQRLOG

I use Fedora 16 x86_64
I use the binary release of CQRLOG "Complete application directory"
I mostly just use CQRLOG to log my contacts and upload them to LoTW, nothing else (so far)
I use CQRLOG 1.3.0

Install dependencies

yum install openssl-devel (needed for LoTW upload support)
yum install hamlib
yum install mysql-server (interesting this binary release of CQRLOG uses distro provided mysql but starts it manually, so you do NOT need to do 'systemctl enable mysqld.service')
yum install trustedqsl

run tqslcert and load your .p12 certificate file
run tqsl and readd your location with the same name as on the other computer

tar up the .config directory for cqrlog from the original computer

tar -cjvf /tmp/cqrlog-backup.tar.bz2 ~/.config/cqrlog/

transfer the cqrlog-backup.tar.bz2 to the new PC and extract it
cd ~/
tar -xvf /path/to/cqrlog-backup.tar.bz2

Download cqrlog_1.3.0_amd64.tar.gz from http://www.cqrlog.com/?q=webfm_send/222/1

cd ~/
tar -xvf /path/to/cqrlog_1.3.0_amd64.tar.gz
run cqrlog as normal

~/cqrlog-1.3.0/usr/bin/cqrlog

All should be good.