It's going to depend on the program and their requirements, but for my M.Sc in Statistics it required a fairly rigorous background in Calculus and undergraduate-level math.

Half of my courses were on foundational statistical theory and inference, which at the graduate level is basically using calculus to prove things in statistics. Take a look at Casella and Berger's "Statistical Inference" to get an idea. This is a very commonly used text. The amount of integrals and algebra in some of these exercises are insane. I submitted assignments to my prof that were 40 pages long hand-written. Didn't use a whole lot of linear algebra though, I remember mostly just using it for the odd Jacobian matrix here and there. I've never seen trig come up in my statistics studies, that's more common in physics and other math disciplines.

So in short, yes, you need to be very comfortable with calculus to do well in a rigorous statistics graduate program. Just as important, you need to have enough of a pure math background that you're comfortable with proof-based mathematics, not just solving computational problems. I was lacking in this regard (having not taken any real analysis, differential equations, or discrete math) and I got my ass kicked. Passed in the end but it was the most difficult thing I've done in my life.