Reconciling the units of pH calculation
$K_a$ is not unitless (in this case). Some equilibrium constants may have their units totally cancel, but the acidity constant has units of concentration:

$$K_a = \dfrac{[\ce{H+}][\ce{A-}]}{[\ce{HA}]}\implies\dfrac{\frac{\text{mol}}{\text{L}}\cdot\frac{\text{mol}}{\text{L}}}{\frac{\text{mol}}{\text{L}}}=\frac{\text{mol}}{\text{L}}$$

I have used moles per liter (molarity) as it is a common concentration unit.

In this case, $K_a$ and $[\ce{H+}]$ both have concentration units, as does $C_a$, so your initial equation $[\ce{H+}]^2 + K_a [\ce{H+}]+ K_a C_a =0$ is valid (at least for units. 

If you recast $K_a$ and $C_a$ in terms of activities, then you need to replace $[\ce{H+}]$ with $\alpha_{\ce{H+}}$ (which is unitless).