Nuts are good because you do want to have some healthy fats in your diet.. there is such a thing as not getting enough fat. But there is always oils, avacodos, olives, seeds, milks, and tofu that have fat in them. You will just have to make sure to include some of them in your diet daily.
As for protein, nuts are one source, but there are many other plant based sources too. You need to eat a variety. Here is a chart
that shows protein content of plant foods (and an egg - bleh).
It sounds like your trainer is not terribly well versed in nutrition, so I would not count on him for that type of advice. In fact I would probably not stick with a trainer like that myself... I would want someone a bit more knowledgeable on plant foods and how best to incorporate them to meet my goals.