No. It's been an ongoing request, but there's no way to do this...other than abandoning a home campaign entirely and only playing as a guest.
On the bright side, no one ever suggested that you play "milkshake" by Kelis in your head every time one of those pops up, so you don't have to cope with that additional annoyance.
PC users can find sanctuary in the addon "No, Thank You!"
"Bah... This is not the way... "
PC EU Tempora Jof, Daggerfall Covenant, Breton, Sorcerer, Vampire: "Being good is hard..."
Jurfi da Nord, Aldmeri Dominion, Nord,Warden, Vampire: "Serving my own needs is easy."
Randal the Second, Ebonheart Pact, Dark Elf, Dragonknight: "I refuse to feel the ex-slaver guilt I have."
Inos sal'Ma'Nefic, Aldmeri Dominion, Imperial, Templar: "Heals by accident count, eh? Eh?"
Desdenada Jof, Ebonheart Pact, High Elf, Sorcerer: "Righteousness is its own edict."
Joffi da Rat, Aldmeri Dominion, Wood Elf, Nightblade: "Bah, this is not the way..."
Indigo sal'Ma'Nefic, Daggerfall Covenant, Redguard, Dragonknight: "For glory!"
PC users can find sanctuary in the addon "No, Thank You!"
This add-on is one of the best things ever developed by the ESO community, giving full control over Cyrodiil and trial-score notifications, along with other great functions. Sadly for the console community, these notifications are non-optional. The only way to stop them is to abandon your home campaign.
Battle not with monsters, lest ye become a monster, and if you gaze into the abyss, the abyss gazes also into you. ~Friedrich Nietzsche
Trial score and other leaderboard notifications can be turned off by going into Settings-->Social-->Turn off leaderboard notifications. Unfortunately the information about the Cyrodil notifications is correct and you'd need the add on.
I feel your pain. These things have gotten me killed a few times soloing pledges. Why they infiltrate into pve i will never know. Its not like you say omg some yellow pleb has picked up a scroll i must abandon what im doing and go retrieve it