Check AppVeyor Webhook at https://github.com/JuliaGraphics/Cairo.jl/settings/hooks. Open Recent Deliveries and see if Webhook is being sent for your change.
If it was sent, what was the response from AppVeyor? If it was good (204), send us exact time (in UTC or with a time zone) Webhook was sent, X-GitHub-Delivery (from webhook header) and Commit ID.
You probably have no enough permissions for the repository. This also may be the reason webhook was not created at all if it was you who created the project on AppVeyor. Or I guessed URL incorrectly (which is not very probable).
Anyway, someone with enough permission should check if GitHub Webhook for AppVeyor was created first.