create table #TestTable (
    PatientName varchar(132),
    ApptNumber int,
    ApptDate datetime
);

insert into #TestTable
values
('John Smith',18374832,'2012-01-05 08:15:00'),
('John Smith',19837289,'2012-01-07 14:30:00'),
('John Smith',19982776,'2012-02-15 09:00:00'),
('John Smith',20091092,'2012-02-27 15:45:00'),
('John Smith',20100939,'2012-02-28 07:25:00'),
('John Smith',20110938,'2012-03-05 16:50:00');

with BaseData as (
select
    PatientName,
    ApptNumber,
    ApptDate,
    rn = row_number() over (partition by PatientName, dateadd(mm, datediff(mm, 0, ApptDate), 0) order by ApptDate asc)
from
    #TestTable
)
select
    PatientName,
    ApptNumber,
    ApptDate
from
    BaseData
where
    rn = 1;

drop table #TestTable;

/*
Given that data the query will only return:
John Smith 18374832 2012-01-05 08:15:00
John Smith 19982776 2012-02-15 09:00:00
John Smith 20110938 2012-03-05 16:50:00
*/