No. You misunderstood that quote, the emergence of writing is far more recent and at different locations. Thus “within 30km of” not at Göbekli Tepe.
“Current archeological evidence in the form of seals, reliefs, steles, lead strips, and wood panels, across almost one-hundred Anatolian sites, including some within 30 km of Göbekli Tepe, dates the emergence of the hieroglyphic script used to write in Luwian to the late 15th century B.C.E.,”
Göbekli Tepe was inhabited ~9500-8000 BCE, so ~6500 years before the writing examples given.
That's the idea - settlements are usually built in favourable areas (e.g. next to rivers) so the same site may be continuously inhabited or repeatedly reinhabited over very long periods of history.
“Current archeological evidence in the form of seals, reliefs, steles, lead strips, and wood panels, across almost one-hundred Anatolian sites, including some within 30 km of Göbekli Tepe, dates the emergence of the hieroglyphic script used to write in Luwian to the late 15th century B.C.E.,”
Göbekli Tepe was inhabited ~9500-8000 BCE, so ~6500 years before the writing examples given.