YES

Problem 1: 

(VAR v_NonEmpty:S x:S)
(RULES
p(f(f(x:S))) -> q(f(g(x:S)))
p(g(g(x:S))) -> q(g(f(x:S)))
q(f(f(x:S))) -> p(f(g(x:S)))
q(g(g(x:S))) -> p(g(f(x:S)))
)

Problem 1: 

Innermost Equivalent Processor:
-> Rules:
 p(f(f(x:S))) -> q(f(g(x:S)))
 p(g(g(x:S))) -> q(g(f(x:S)))
 q(f(f(x:S))) -> p(f(g(x:S)))
 q(g(g(x:S))) -> p(g(f(x:S)))
-> The term rewriting system is non-overlaping or locally confluent overlay system. Therefore, innermost termination implies termination.
 

Problem 1: 

Dependency Pairs Processor:
-> Pairs:
 Empty
-> Rules:
 p(f(f(x:S))) -> q(f(g(x:S)))
 p(g(g(x:S))) -> q(g(f(x:S)))
 q(f(f(x:S))) -> p(f(g(x:S)))
 q(g(g(x:S))) -> p(g(f(x:S)))

Problem 1: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 p(f(f(x:S))) -> q(f(g(x:S)))
 p(g(g(x:S))) -> q(g(f(x:S)))
 q(f(f(x:S))) -> p(f(g(x:S)))
 q(g(g(x:S))) -> p(g(f(x:S)))
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.