I have been eating about a quart of yogurt a day for a week.
Could this cause weight gain from all of the protein?
Calorically, it fits into my diet.
But, I was wondering, would all of the protein lead to weight gain?
A calorie is a calorie. If you're not exceeding the amount of calories than what you burn then you will not gain any weight.
I find it highly inplausible that an excess of yoghurt protein could cause you to gain weight. While keeping an eye on your macronutrient intake is important, it does not have any effect on weight loss over a long period of time (studies have shown short-term advantages to eating more/less protein/fat/carbs, but over months and years, there is no difference when it comes to weight loss, as far as I'm aware).
If you are eating flavoured yoghurt, then the extra sugar could affect your insulin levels and hinder fat burning, and other added ingredients may have some effect.
However, the most important thing to remember is that your weight is not a result of a single factor, (such as eating too much yoghurt), it is a result of your overall lifestyle. In other words, even if eating too much yoghurt could make you gain weight, it would not be a very significant amount, especially over a period of one week. If you gain weight, it is because of your overall diet and lifestyle, not because of one type of food.
If it fits into your calorie count you shouldn't show a gain, but I have a feeling you may get sick of eating all that yogurt..or at least bored. Maybe try alternating something else.
Also are you eating full fat yogurt? I recommend you do if you have high protein intake--you may not be able to eat as much calorie wise, but I personally find it more satisfying in the long run. I'd also recommend Greek yogurt over the other stuff.