Dimensional analysis for the qdf/quantile function corresponding to the pdf/CDF for the size distribution of income
Your random variable is $X$ = "Yearly Income per capita measured in dollars". So $X$ is a real-valued function. Its domain is usually left unspecified, while its range is measured in "dollars per person-year", as the OP writes it.

The density and the distribution (pdf/cdf) functions of $X$ ,say $f_X,F_X$ have as their domain the range of $X$. So the domain of $f_X,F_X$ is a set that carries the unit of measurement "dollars per person-year". 

Being strict in dimensional analysis, indeed we could say that the density must then be measured in "person-years per dollar" (irrespective of whether this is meaningful or not). 

So the expected value that is the integral $\int_0^{\infty} x f_X(x) dx$ keeps having "dollars" as a measurement unit (I just assumed $X$ is non-negative), applying the rule the OP states (the one unit is the reciprocal of the other, and we are left with the third which is again "dollars per person-year"). 

Moreover the integral that is the cdf $F_x(x) = \int_0^x f_X(s) ds$ has a range that comes out "dimensionless", since probability is a relative measure.

(Digression: Note that by definition, the distribution function of a random variable measures the probability of the random variable being smaller or equal than a threshold, $F_X(x): = Prob(X \leq x)$. Additional assumptions are required to "tranlsate" this probability as representing also "income shares", meaning I guess "percentage of population with per capita income below or equal to $x$")

Turning to the Quantile function, it is defined as the inverse of the distribution function (generalized inverse to cover all cases, but let's keep it simple). Being the inverse, it uses as its domain the range of $F_X(x)$, so the domain of $Q(p)$ carries "probability" as unit of measurement, while its range is the domain of $F_X$. And the domain of $F_x$ is the range of $X$: the Quantile function has the same range as $X$ and so it is measured in "dollars per person-year" too (i.e. its range caries the "dollars per person-year" unit of measurement).

it follows that the derivative of $Q(p)$", $q(p) = dQ(p)/dp$ has the same units of measurement as $Q(p)$, while the integral $\int_0^1 pq(p)dp$ is measured in "dollars per person-year" also.