Termination w.r.t. Q of the given QTRS could be proven:

(0) Obligation:

Q restricted rewrite system:
The TRS R consists of the following rules:

f(x, nil) → g(nil, x)
f(x, g(y, z)) → g(f(x, y), z)
++(x, nil) → x
++(x, g(y, z)) → g(++(x, y), z)
null(nil) → true
null(g(x, y)) → false
mem(nil, y) → false
mem(g(x, y), z) → or(=(y, z), mem(x, z))
mem(x, max(x)) → not(null(x))
max(g(g(nil, x), y)) → max'(x, y)
max(g(g(g(x, y), z), u)) → max'(max(g(g(x, y), z)), u)

Q is empty.

(1) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Lexicographic path order with status [LPO].
Quasi-Precedence:

[f₂, nil] > [g₂, ++₂, false] > [mem₂, =₂] > or₂
[f₂, nil] > [g₂, ++₂, false] > max₁ > null₁ > or₂
[f₂, nil] > [g₂, ++₂, false] > max₁ > not₁ > or₂
[f₂, nil] > [g₂, ++₂, false] > max'₂ > or₂
[f₂, nil] > true > or₂
u > [g₂, ++₂, false] > [mem₂, =₂] > or₂
u > [g₂, ++₂, false] > max₁ > null₁ > or₂
u > [g₂, ++₂, false] > max₁ > not₁ > or₂
u > [g₂, ++₂, false] > max'₂ > or₂

Status:

f₂: [2,1]
nil: []
g₂: [2,1]
++₂: [2,1]
null₁: [1]
true: []
false: []
mem₂: [2,1]
or₂: [2,1]
=₂: [2,1]
max₁: [1]
not₁: [1]
max'₂: [2,1]
u: []

With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

f(x, nil) → g(nil, x)
f(x, g(y, z)) → g(f(x, y), z)
++(x, nil) → x
++(x, g(y, z)) → g(++(x, y), z)
null(nil) → true
null(g(x, y)) → false
mem(nil, y) → false
mem(g(x, y), z) → or(=(y, z), mem(x, z))
mem(x, max(x)) → not(null(x))
max(g(g(nil, x), y)) → max'(x, y)
max(g(g(g(x, y), z), u)) → max'(max(g(g(x, y), z)), u)

(2) Obligation:

Q restricted rewrite system:
R is empty.
Q is empty.

(3) RisEmptyProof (EQUIVALENT transformation)

The TRS R is empty. Hence, termination is trivially proven.

(4) TRUE