At the very least it is odd the only intelligence agencies we've heard of are directly connected to Starfleet.
Not really, our PoV has been universally that of Starfleet personnel, so it makes sense that we haven't seen any civilian intelligence agency (though it's possible that Federation Security, like the FBI, has both security and intelligence functions for the Federation as a whole, whereas wholly local security and/or intelligence issues are the purview of civilian and/or military authorities of that planet or territory.
Starfleet Intelligence being the main intelligence division of Starfleet and Section 31 supposedly originating from the Starfleet Charter.
Yup.
Doesn't the Federation government have its own civilian intelligence agency?
See above.