It doesn't really "integrate" but you can set up a pjirc client and pass the boardnick (if logged in) via the link.
You should use a prefix then (like wursts-<nick>) so ppl don't collide with already logged in users.
Just an easy solution
Don't know about a bot with actual mysql support, but there once was a php driven bot out there, sql access should be only a few lines away then.
[EDIT]
Just checked on that phpbot issue due to boredom and found several bots fully written in php cappable to use extra scripts. If you still think you have to write your own bot I stumbled over this:
http://www-128.ibm.com/developerworks/l ... ml?ca=drs-[/EDIT]
Personally I don't like quakenet (due to weird personal, not explainable reasons) but maybe I'll pass by some day ...