I do this on my site (kinda). Here is the code I use. I think I changed it correctly from what I had to what you want (of course, use on your test site first!). You will have to put it where you want with the correct html around it, and don't put it inside any other form cuz that won't work.
HTML Code:
<form action="postings.php?do=domovethread&t=$threadid" method="post" name="formname">
<input type="hidden" name="s" value="$session[sessionhash]" />
<input type="hidden" name="securitytoken" value="$bbuserinfo[securitytoken]" />
<input type="hidden" name="t" value="$threadid" />
<input type="hidden" name="do" value="domovethread" />
<input type="hidden" name="title" value="$threadinfo[title]" />
<input type="hidden" name="redirect" value="none" />
<input type="hidden" name="redirecttitle" value="$threadinfo[title]" />
<input type="hidden" name="destforumid" value="xx" />
<input type="submit" class="button" value="Move Show" />
</form>
Replace
xx with the forumid and each form should have a unique name (I put formname as an example in this one). You will need a form for each Quick Move button.