Solution:

Identify the plane with the complex plane, so a move sends $z$ to $\omega z + 10$ with $\omega = e^{i\pi/4}.$ Starting from $z_0 = 5$ and iterating, $$z_{150} = 5\omega^{150} + 10\left(\omega^{149} + \omega^{148} + \cdots + \omega + 1\right).$$

Since $\omega^8 = 1$ and $150 = 8 \cdot 18 + 6,$ we get $\omega^{150} = \omega^6 = -i.$ In the geometric sum, every block of $8$ consecutive powers adds to $0,$ so the $150$ terms reduce to $1 + \omega + \cdots + \omega^5 = -\omega^6 - \omega^7 = i - \left(\frac{\sqrt{2}}{2} - \frac{\sqrt{2}}{2}i\right).$ Therefore $$z_{150} = -5i + 10\left(-\frac{\sqrt{2}}{2} + \left(1 + \frac{\sqrt{2}}{2}\right)i\right) = -5\sqrt{2} + \left(5 + 5\sqrt{2}\right)i.$$

Thus $|p| + |q| = 5\sqrt{2} + 5 + 5\sqrt{2} = 5 + 10\sqrt{2} \approx 19.14,$ and the greatest integer less than or equal to this is $19.$