Is there a UserInputService equivalent to Mouse.Idle?

Unfortunately the users mouse in UserInputService is a constant event so it will not idle / have an end state.

Although you could use InputChanged to detect if the Mouse has been moved like so:

UIS = game:GetService('UserInputService')

mouseIdle = false

mouseIdleDelay = 10 -- in seconds

UIS.InputChanged:Connect(Input)
    if Input.UserInputType == Enum.UserInputType.MouseMovement then
        mouseIdle = false
    end
end)

spawn(function()
    while wait(mouseIdleDelay) do
        mouseIdle = true
    end
end)

If you have any other questions feel free to let me know.