How much does a cup of hazelnuts weigh?

Well 1.5 cups is volumetric so the weight will depend on how tightly packed that cup is, if they are chopped, etc. On average though, 1.5 cups of chopped hazelnuts is about 6.08 oz (172 g) according to the USDA's averages. You can check the USDA's National Nutrient Database page for hazelnuts if you want to see estimated weights for 1.5 cups of whole, chopped, or ground hazelnuts.

The first place you can look at when you have such a question is a cooking converter. There are several such online, I use mostly the one at http://www.convert-me.com/en/convert/cooking, but I don't know if it is better or worse than the competition. By the way, it lists 1.5 cups of whole hazelnuts as 255 grams.

Another thing you can do is to find the correct ratio from a known good recipe (the best sources will list ingredients by weight) and use that one. I was going to look for a gianduja ratio for you when I took a better look at your recipe.

While it calls your ice cream gianduja ice cream, it does not contain or use gianduja. Gianduja is two things - either a paste made from hazelnuts and powdered sugar (analog to raw marzipan), or this paste mixed with cooking chocolate or pure cocoa butter. Your recipe uses milk infused with hazelnut, which is a different thing entirely. I suspect that the exact ratio is not too important for such an infusion, but if you want to know an optimal infusion ratio, look for other recipes which use a nut-milk infusion but are better quality, use them to prepare the infusion, then continue with your recipe as it is.