Does pH affect Michaelis constant?
From the derivation of Michaelis-Menten kinetics you can see that:

$$K_m=\frac{k_f + k_{cat}}{k_r}$$

Where $k_f$ and $k_r$ are binding and unbinding rate constants (for Enzyme-Substrate binding), respectively, and $k_{cat}$ is the turnover number. This is for the Quasi-Steady-State approximation (QSSA). For the equilibrium approximation:

$$K_m=\frac{k_f}{k_r}$$ which is same as the association constant.

In both the cases pH can affect the rate of binding and unbinding by affecting the affinity between the enzyme and the substrate. For example lets assume that the substrate binding site is negatively charged. Low pH would increase the electrostatic potential of the substrate binding site towards zero by affecting the ionization of the functional groups. 

pH can also have indirect effects on the substrate binding site because it can modify the overall structure of the protein.

Many enzyme catalysed reactions involve acid-base catalysis i.e. there is a transfer of proton. In reactions like these, pH can affect $k_{cat}$.  


  If pH does change Km, is this how I determine the Km value of a
  different pH value if the Km value at another pH is already known?


You can do that by making a Lineweaver-Burk plot for the changed pH.