Use the logarithm property for division.

The property states that $\log_b \left(\frac{X}{Y}\right) = \log_b X - \log_b Y$.
Applying this to the expression $\log_4 \left(\frac{m}{4}\right)$:

$$\log_4 \left(\frac{m}{4}\right) = \log_4 m - \log_4 4$$

We are given that $\log_4 m = 12$.
Also, $\log_b b = 1$, so $\log_4 4 = 1$.
Substitute these values into the equation:

$$\log_4 \left(\frac{m}{4}\right) = 12 - 1$$ $$\log_4 \left(\frac{m}{4}\right) = 11$$

The value of $\log_4 (m/4)$ is $\boxed{11}$.