london – Will my Oyster card work on Farringdon to Brighton Thameslink rail?

I assume you mean the Brighton which is on the south coast of England.

If so, no.

You can start the travel on your Oyster Card, but when you reach the end of the validity area you will have to leave the train and buy a new ticket.
It will be easier to buy a ticket from your starting station all the way to Brighton, which will be easier and likely be cheaper.