It is unfortunate that the usual poor journalism labels the growth of the black hole as "inexplicable" and then further down in the article refers to some possible explanations.
The basic problem is a growth timescale one. Radiation pressure introduces a negative feedback, such that there is a "theoretical" maximum for spherical accretion called the Eddington limit, which occurs when the quasar is radiating at its Eddington luminosity. The shortest growth timescale is achieved if the efficiency with which mass is converted to luminosity is low; but if it's too low we wouldn't see the quasar at all. This is the crux of the problem. You can look at this Physics SE answer for some more of the details.
The thing is there are ways and means by which this limit can be exceeded - non spherical accretion for one - so there are lots of ideas about how this can be achieved. Another possibility is that you start off with a seed black hole that is pretty big to begin with, perhaps as a result of a merger. Or the quasar could have been less efficient in the past and is more efficient now, which is why we can see it.
Is there enough matter? Well, yes, galaxies have masses that can be much bigger than the mass of this black hole. They are rare, but of course so are > 1 billion solar-mass black holes, and these tend to be the only ones that we can see at distances of >10 billion light years.
One way of assessing the feasibility would be just to ask what a freefall timescale would be. If you have say $10^{11}\ M_{\odot}$ in a sphere of radius 10 kpc (I am just using typical sorts of numbers for a big galaxy), then the average density is $5\times 10^{-22} kg/m^{3}$ and has a freefall time $\sim (G\rho)^{-1/2}$ of 200 million years. Of course there are other problems, like shedding angular momentum, but it looks like this timescale is short enough for gravity to do its thing (in the absence of radiation pressure).
Of course the short answer to your question is that yes, there must be enough time, because this is just the latest in a population of such objects. We know that quasars with supermassive black holes have formed within a billion years after the big bang.
