I was about to answer "No", but re-reading your question I suspect that what you meant to ask was
"Would it be possible to use the navigation software on the Nokia N810 to receive and use data from a remote GPS receiver".

If so, then yes, in principle at least. But you would need a way of transmitting the data. The GPS receiver on the dog may be combined with a radio transmitter of some sort, e.g. VHF or something. Or even a GSM phone (they use those on sheep). So, what you then need on your N810 is a way of receiving the data. If it's via VHF or some other kind of radio then you would need a receiver, if the dog GPS unit sends data via a GSM phone (which could be via GPRS or even the occasional SMS) then you would need a piece of software to retrieve the data from wherever it gets stored. And finally a little piece of software to feed NMEA data to gpsd, the GPS daemon on your N810.