Before making any decisions, look carefully at the power consumption of the collar (is this one of those LEDs flashing at night sort so you can see which bushes the dog has rushed in to), and how long it has to run for between charges.
You'll need to fit in a big enough battery for that, then make further room for wireless charging circuitry. The simplest wireles charging would be done with a big coil on the charging device, which is set up as the inductor in an LC parallel tank circuit. The device then has it's own LC parallel tank tuned for the same frequency, potentially using a purchased inductor as its coil (10mH coils with diameters around 1cm, lengths around 2cm... can work so long as not shielded), this picks up an AC signal which you need to rectify to DC with which to then power a battery charging circuit (designed for whichever battery chemistry you're battery uses). There will always be more losses of power during charging, due to the intermediate steps of DC-->AC in charger's LC tank-->induced AC in device's LC tank-->rectified DC than there would in just supplying DC to the battery charger circuit in the first place.
These four videos, and I know video is often less helpful then a written article with images, give some useful info. They were ones I found in my bookmarks history, having read up on LC induction for another reason a while back (I built an absolute angle inductive resolver).