For callable bonds we can use the effective duration to approximate the modified duration, since the future interest rates will affect the expected cash flows. For convertible bonds the underlying of the embedded option does not depend of the interest rate levels but the underlying also affects the (possible early) redemption.
Unfortunately, convertible bonds are quite complex so you don't have simple formulas or approaches as with vanilla bonds. However, this does not mean you are powerless. You can follow different approaches, for instance:
1) If you have a pricing model for the convertible, you can use the finite difference method (or other techniques) to get any sensitivity. Note that this is the same thing you do with the Effective Duration however here is more complex as what you are capturing is the change in the convertible price with respect to rates. In my opinion, this is the most correct approach, however, you have quite a bit of model risk (garbage in - garbage out). Also notice that if you see the convertible as a function of three elements - bond, equity and option - what you are really capturing here is not the pure bond duration but rather a function of the impact that rates have on all of them (which is the right thing you should be capturing)
2) You can completely ignore the optionality and calculate the duration of an equivalent vanilla bond using the bond floor of the convertible as its price. This approach is incorrect but if two bonds are in the same delta bucket (i.e. similar deltas) then this approach can be considered an acceptable rule of thumb for comparison. However, forget hedging or risk management here as everything would be incorrect
3) You can be creative about it. For instance, one approach I frequently use (and I admit that it is flawed) is to take the vanilla duration of 2) and multiply it by (1 - square root of the delta of the convertible). This way if the converts is a lot in the money the duration with respect to the bond is zero (makes sense because you don't care about the price of the bond). Conversely, if the delta is very low than essentially it is a busted convert so it should be very correlated to a vanilla bond. In the middle, it gets a bit and a bit but I like to give more weight to the equity. This is so as in convertibles the duration becomes a rounding error as the option gets in the money as the optionality tends to dominate the variance of the convertible and therefore people care much more about the delta, IV and so on rather than duration... I hope this helps