You would start by writing code to numerically integrate your DE using ode45. You have a simple 1st order ODE. There is an example in the ode45 doc showing how to do a 1st order ODE. Start with that and modify it with your equation using the time span of your data.
Once you have that working, wrap your ode45 stuff inside of a function where you pass in q to the function.
Then ask yourself how you would use an optimizer on this function to minimize the difference between what your function produces for a given q to what your measurements are.
