**It's possible, but unless you have an _incredible_ appetite, there's little evidence that it would have any effect.**
Large amounts1 of tryptophan have been shown1 2 to reduce sleep latency (the time it takes to fall asleep), however the data is less clear on whether it causes drowsiness or has any effect on the length or quality of sleep.
Though tryptophan is one of the 9 essential amino acids and, as such, is found in any source of complete protein (i.e. eggs, soy, dairy, or any meat) and several incomplete sources (potatoes, rice, wheat, etc.). Turkey meat doesn't contain an especially great amount of it (about 0.24g/100g), being comparable to chicken or beef. Egg whites, soy, and cod contain far more typtophan, yet those foods seem to have avoided any association with sleepiness.
1. More than 1 gram, which would require over 400 grams (nearly 1 pound) of turkey meat to get.