No, typically you just need the packets to be sent to your NIC and have it capture passively. Chances are that you're having a problem getting the STP packets delivered to your NIC in the first place.
If you're capturing by mirroring traffic using a SPAN (or other kind of "monitoring" port), many switches will not forward the STP traffic from one port to another. Depending on your switch there may be a way to configure it to do this, but most likely you won't be able to capture STP traffic using a monitor port. You'd have to use something to let you tap the physical connection between the switches. There's hardware that will let you do that. Heck, inserting an ethernet hub on the trunk line and attaching your system to that would do the trick.