Sure, if your L/C meter is designed to work at those capacitances. The capacitance of a gamma rod isn't much -- it's probably measured in pF. The lowest range of the inexpensive L/C meter I have from eBay is 200pF, which is fine for most of the capacitors in my parts drawer, but not so great for RF.
Also keep in mind: if you are making connections with long alligator clips, these are going to have inductance and capacitance in addition to the gamma rod. Depending on how well the clips chomp down they may add significant resistance, too. Unless you take some very special care to keep those leads short or compensate for them, you are going to introduce so much stray reactance with the leads that any measurement you obtain won't be of much use.
In practice, anything designed to measure the reactances typical at RF stops being called an "L/C meter" and starts being called an "antenna analyzer" or a "network analyzer". So, while it certainly is possible to measure the capacitance of a gamma match, it's probably not possible to do it with something marketed as an "L/C meter".