YES

TRS:

  p(f(f(x))) -> q(f(g(x)))
  p(g(g(x))) -> q(g(f(x)))
  q(f(f(x))) -> p(f(g(x)))
  q(g(g(x))) -> p(g(f(x)))

linear polynomial interpretations on N:

  p_A(x1) = 0
  p#_A(x1) = x1 + 1
  f_A(x1) = x1 + 3
  f#_A(x1) = 0
  q_A(x1) = 0
  q#_A(x1) = x1
  g_A(x1) = 1
  g#_A(x1) = 3

precedence:

  g > q > p > f