No, we are not inside an active galactic nucleus (AGN). An active galactic nucleus refers to the extremely bright and energetic core region of a galaxy that is believed to be powered by accretion of matter onto a supermassive black hole at its center. AGNs are found at the centers of galaxies, but they are not structures in which stars or planetary systems, including our own, reside.
Our solar system is located in the Milky Way galaxy, which is a large spiral galaxy. The center of our galaxy contains a supermassive black hole known as Sagittarius A* (Sgr A*), but it is currently in a quiescent or inactive state, unlike an AGN. The activity of AGNs is characterized by the presence of intense radiation, jets, and other energetic phenomena, which are not observed in the immediate vicinity of our solar system.
While the central regions of galaxies can have a variety of active and dynamic processes, including AGNs, our solar system is situated far from the core of the Milky Way and does not experience the conditions associated with an AGN.