that should be possible. Just find an dogbark mp3 and send it to the speaker should be quiet easy.
I do something similar, but with Android tablet in one room, and with a speaker attached to the HA server in another room. I’m sending an mp3 of a train station announce ding-dong before a voice message from TTS.
Just I do not use Google speakers, so I can’t give you exact steps. But is is for sure doable 