It doesn't suprise me at all that you can't find a replacement, apart from the hard drive and RAM, laptop computer parts tend to be proprietary, unlike computer desktop motherboards which tend to be generic. Also Toshiba wouldn't sell laptop motherboards to the end user directly, they would have to be ordered through a repair centre. Your best bet is to look on eBay or Amazon or Aliexpress for a working second hand board or send the laptop to a Toshiba service centre
'Socket' BGA-1364 isn't a socket in the desktop sense where the CPU is mechanically held in and can be removed. It's rather an interface where the CPU package and motherboard are soldered together.
You'd need specialist equipment to separate your CPU from the current board and re-solder it to a new motherboard. The CPU & motherboard is generally treated as a single component and replaced together.
So you'd need to be talking to dedicated laptop repair places if you want to try and keep the current CPU, and even then you may have trouble sourcing a motherboard without CPU. And the 4710HQ was just about the cheapest CPU to use BGA-1364 so you won't even be able to scavenge a motherboard with a cheaper CPU attached.