Absolutely a month is reasonable. I was TOTALLY surprised at how quickly I saw definition when I started weight training. I had told myself that, like losing weight, I shouldn't expect to see results too quickly and so really given myself some low expectations. But I saw changes in my arms especially very quickly.
Now, I will say that I had already lost about 40ish lbs when I started weight training, so there was less fat padding my muscle, but even so, I was very happy with the changes I saw the first month. And it's really motivating, too!