No. The beginning of your upkeep step is a single event wherein anything that will trigger from that does so, and then the opportunity is past. Frenzied Fugue's ability has to resolve before you gain control of Damia, and by that point it's too late for Damia to also trigger for you.
You would have to have more than one upkeep step (likely via Paradox Haze) in order to do what you want.
Is it possible to control Damia at the beginning of my upkeep and trigger her ability?
Shu Yun, the Silent Tempest WUR Voltron Control
Temmet, Vizier of Naktamun WU Unblockable Mirror Trickery
Ra's al Ghul (Sidar Kondo) and Face-Down Ninjas
Brudiclad, Token Engineer
Vaevictis (VV2) the Dire Lantern
Rona, Disciple of Gix
Tiana the Auror
Hallar
Ulrich the Politician
Zur the Rebel
Scorpion, Locust, Scarab, Egyptian Gods
O-Kagachi, Mathas, Mairsil
"Non-Tribal" Tribal Generals, Eggs
You would have to have more than one upkeep step (likely via Paradox Haze) in order to do what you want.
Shu Yun, the Silent Tempest WUR Voltron Control
Temmet, Vizier of Naktamun WU Unblockable Mirror Trickery
Ra's al Ghul (Sidar Kondo) and Face-Down Ninjas
Brudiclad, Token Engineer
Vaevictis (VV2) the Dire Lantern
Rona, Disciple of Gix
Tiana the Auror
Hallar
Ulrich the Politician
Zur the Rebel
Scorpion, Locust, Scarab, Egyptian Gods
O-Kagachi, Mathas, Mairsil
"Non-Tribal" Tribal Generals, Eggs