Learn German: Der Vertreter der Regierung liest den Vertrag. - The representative of the government is reading the treaty.

You're right, this sentence just uses a regular verb with a direct object (liest den Vertrag). However, you will frequently see Vertrag in the NVV unter Vertrag stehen (to be under contract) and jemanden unter Vertrag nehmen (to sign someone to a contract).

Why is it der Regierung if Regierung is a feminine noun?

This is a use of the genitive case to show possession or belonging ("of the government"). In the genitive case, the feminine definite article die changes to der.

Does Vertreter only mean "representative" in a political or legal sense?

No, it's a very versatile word! While it means "representative" here, it is also the standard word for a "sales representative" or a "substitute" (such as a substitute teacher). It comes from the verb vertreten, meaning to represent or to stand in for someone.

The English translation says "treaty". Doesn't Vertrag usually mean "contract"?

It means both! Der Vertrag is the general word for a legally binding agreement, whether that's an everyday cell phone contract (der Handyvertrag) or an international treaty between governments. For international treaties, you will also often see the word das Abkommen.