This is not a sentence. A sentence has (most often) a subject and always a predicate. But in this group of words there is no predicate. The core of a predicate is a verb, but although the word »gibt« is a verb, it still is not the sentence's predicate (or a part of it).
A full sentence might be:
Ein Held, der sich heute bei euch die Ehre gibt, trägt einen grünen Hut.
A hero who pays his respects to you today wears a green hat.
This sentence has these parts:
- ein Held, der sich heute bei euch die Ehre gibt
These 10 words together are the subject of the sentence, and this subject occupies position 1 of the sentence.
This is the predicate, and it consists of only one word which is a verb. It occupies position 2 of the sentence. Position 2 is the place where you always will find a verb in any correct German statement (not in commands or questions).
- einen grünen Hut
The verb trägt doesn't need just a subject, but also an object in accusative case, and these 3 words are together the accusative object of this sentence.
So, let's have a deeper look on the subject, which is identical to the phrase from your question:
- ein Held
This is a nominal group in nominative case, and it consists of a determiner (here: an article) and the core of the nominal group, which is a noun (this is why we call it a nominal group)
- der sich heute bei euch die Ehre gibt
This a right attribute of the nominal group which contains some more details about what is described by the nominal group itself. And this right attribute is a "relative clause" (In German: »Relativsatz«) In German we use the term Satz for full sentences but also for clauses, which might be confusing. Just know, that a »Relativsatz« (relative clause) is not a »ganzer Satz« (full sentence)
A German relative clause always starts with a relative pronoun (here: »der«) and always ends with a verb (here: »gibt«). But this verb belongs only to the relative clause. It has no connection to the other parts of the sentence outside the relative clause, so it can't be part of the full sentence's predicate.
But there is a way to convert the phrase in discussion into a full sentence, and then the former relative clause's verb will become the full sentences finite verb and therefore must be located at position 2:
Ein Held gibt sich heute bei euch die Ehre.
A hero pays his respects to you today.
Now we have a full sentence, and it has it's verb »gibt« where it belongs to: at position 2.