Both versions are possible, but they have different nuances.
Welches ist die zuständige Behörde?
The welches here is entirely generic. It can be substituted with everything, which is especially helpful if we don’t know what we are really asking for. It’s like saying ‘I know someone is the appropriate authority, but I have no clue who they actually are.’
Welche ist die zuständige Behörde?
This is a modification of the first sentence but knowing what kind of a thing we are looking for, and implicitly expecting an answer that has feminine genus.
Welche Behörde ist zuständig?
This sounds more like ‘we have a list of possible authorities; which one should I ask?’ It sounds a tad more closed to me than the open-ended welches ist …
Essentially, all three are understood and all three are more or less equally valid. More detail can be found on canoo.net.