Can you provide an explanation of how a "quartile" is calculated when
measuring a person's salary against others?  Specifically, if I want
to hire an employee and pay an amount which would place them at the 55
percent quartile compared to peers, how would it be calculated?  I
have "peer" salaries of: $26,300 / $27,948 / $32,350 / $34,490 /
$39,500 / and $45,700.

55% quartile is a tautology.  To establish quartiles, rank your
numbers in order.  The number 1/4 of the way along the list is the
first quartile, the second, the second quartile, etc.  For your
numbers - only six of them, the statistic is essentially meaningless. 
Given a larger set, you could however hace a 55th percentile -
calculated similarly.

"A percentile is a value on a scale of one hundred that indicates the
percent of a distribution that is equal to or below it. In general, a
percentile greater than 75 is considered above normal, and stated
another way - 75 percent of the surveyed population is paid below you
- and only 25 percent are paid above you, 1) a percentile between 25
and 75 is considered normal , 2) a percentile at 50 percent is equal
to the median, 3) a percentile less than 25 is considered below
normal, or alternatively, 25 percent of the surveyed population is
paid below you and 75 percent are paid above you."
Definition from the Economic Research Institution
http://www.eri-salary-survey.com/salary_termsIII.html

So, if you have only one person being paid each of your cited
salaries, i.e. six employees in total, then all you can really say is
that $32,350 gives you the 50th percentile and $34,490 gives you the
67th percentile.

Perhaps this might work for you:

Take the mean of the peers' salaries - in this case about $34,381, and
say this is "about" the 50th percentile.  Then add 10% to give a "sort
of" 55th percentile.  That would be $37,819.

Just a thought.

Perhaps the word you're looking for is 'quantile'?

All good answers. Quartile is not very useful in the case of 6
employees, perhaps not even 600.   Neil