The GBM's distribution at any given point is $$\mathcal N(\mu t,t\sigma^2)$$
Hence the 90th percentile would be $\Phi^{-1}(\mu t,t\sigma^2,0.9)$. It is certainly concave.

Now, your curve is an empirical 90th percentile, and it's a random curve as such. It will not have a derivative at any point in an ordinary sense, like @whuber mentioned. Hence, in an ordinary sense, i.e. second derivative's sign, it cannot be concave or convex. You'd have to define some measure of convexity for a random curve that doesn't have derivatives in order to judge whether your curve is concave or convex.