Count the number of occurences in a query result
I think you just need a window count():

COUNT(*) OVER (PARTITION BY AuctioneerSurname, AuctioneerName)


The query with the added computation - and using aliases for the long table names:

select 
    app.AppointmentDate,  app.AppointmentTime,
    auc.AuctioneerName,  auc.AuctioneerSurname, 
    buy.BuyerName,  buy.BuyerSurname,
    AuctioneerCount = count(*) over
                         (partition by auc.AuctioneerSurname, auc.AuctioneerName)
from 
    Appointment as app
    inner join Auctioneer as auc 
        on app.AuctioneerId = auc.AuctioneerId  
    inner join Buyer as buy 
        on app.BuyerId = buy.BuyerId  
where
    app.AppointmentDate between '2017-01-01' and '2017-12-31' ;