The quotient of two negative integers results in an integer. how does the value of the quotient compared to the value of the original two integers?

First, the quotient is positive (since the multiplication or division of any two negative numbers is positive). Second, the relationship between the first two numbers (let's describe them as "a" and "b", these are just placeholders) and the quotient (we'll describe this as "c") is simply multiplication and division, as shown below:
if
          a/b =c
then
          a = b*c
          b = a/c