Understand the properties of the angles in STR.

In $\triangle STR$, the sum of angles is $180^\circ$.
We are given that $\angle STR = 44^\circ$.
RT is the bisector of $\angle R$, so $\angle SRT = \frac{1}{2} \angle R$.
ST is the bisector of $\angle S$, so $\angle TSR = \frac{1}{2} \angle S$.
Therefore, we can write the sum of angles in $\triangle STR$ as:

$$\angle STR + \angle SRT + \angle TSR = 180^\circ$$ $$44^\circ + \frac{1}{2} \angle R + \frac{1}{2} \angle S = 180^\circ$$

From the equation in Step 1:

$$\frac{1}{2} (\angle R + \angle S) = 180^\circ - 44^\circ$$ $$\frac{1}{2} (\angle R + \angle S) = 136^\circ$$ $$\angle R + \angle S = 2 \times 136^\circ$$ $$\angle R + \angle S = 272^\circ$$

The sum of interior angles in any quadrilateral is $360^\circ$.

$$\angle P + \angle Q + \angle R + \angle S = 360^\circ$$

Substitute the value of $(\angle R + \angle S)$ from Step 2:

$$\angle P + \angle Q + 272^\circ = 360^\circ$$ $$\angle P + \angle Q = 360^\circ - 272^\circ$$ $$\angle P + \angle Q = 88^\circ$$

The ratio of $\angle P$ to $\angle Q$ is $5:6$.
Let $\angle P = 5x$ and $\angle Q = 6x$.
Substitute these into the equation from Step 3:

$$5x + 6x = 88^\circ$$ $$11x = 88^\circ$$ $$x = \frac{88^\circ}{11}$$ $$x = 8^\circ$$

Now, calculate the measures of $\angle P$ and $\angle Q$:

$$\angle P = 5x = 5 \times 8^\circ = 40^\circ$$ $$\angle Q = 6x = 6 \times 8^\circ = 48^\circ$$

Difference = $|\angle Q - \angle P|$
Difference = $|48^\circ - 40^\circ|$
Difference = $8^\circ$

The difference between the measures of $\angle P$ and $\angle Q$ is $\boxed{8^\circ}$.