In the sentence below, is nice a subject complement, verb complement, or both?
She seems nice.
The most traditional terminology is that verbs like seem are linking verbs that link the subject (or object) with their corresponding subject/object complements. A subject/object complement is also known as a predicative.
Predicatives are also complements to the linking verb, though: they are part of the predicate in the sentence. (With the verb be, in fact, they are the predicate, and the verb itself is just an empty placeholder that has to be there for the sentence to be grammatical.)
So the simple answer to your question is both. A subject complement or subject predicative is automatically a complement to the main verb in the clause, if there is one.
In the sentence below, there are all kinds of names for what nice is.
But the sentence above has been done things to. An equivalent sentence is
from which the first sentence is derived by To-be-Deletion.
In other words, there is an infinitive complement here.
And (be) nice is its predicate adjective.
What you want to call nice after to be is deleted is up to you.
I won't go into A-Raising here, but seem is a prototypic A-Raising verb,
since what actually seems in this sentence is in fact not her, and not nice, but rather
that she is nice.
That she is nice is also a possible complement clause with seem, although it requires Extraposition.
And it means the same as the other sentences.