Rewrite the equation as 85^m = n^4 + 4. Then the right side factors into (n^2-2n+2) * (n^2+2n+2). Each of these factors is one more than a square, so 85^m = [(n-1)^2 + 1] * [(n+1)^2 + 1)].

If m=1 then 85^m = 85^1 = 5*17 = [2^2 + 1] * [4^2 + 1], which implies n=3.

For m>1 I can't see a way to split factors 5^m * 17^m in a way such that the factorization is the product of two numbers of the form x^2+1.