Our Luna is nearly 2 1/2 and has never done this behavior before, now when we go out for walks after a little while she finds a stick to run with and will stop ahead of us and chew it, we have tried ignoring it and hoping it stops but it hasn't, we have tried luring with tasty treats but she will only chimp on it harder and quickly, we never play with sticks with her or throw them so not sure why she has suddenly started to do this .
I think some Labs are part Woodchuck. All of our Labs have occasionally chewed up sticks. It has never caused a problem, other than being a bit messy when they do it in the house.