The most used equivalent for "apropos" would be "Hablando de..." (Speaking of...).
Apropos of cars, I bought one! -> Hablando de coches, ¡me he comprado uno!
About "apropos of nothing", it would depend on context, as @Gustavson already told. My suggestions:
Por nada en especial (for no special reason)
Sólo por curiosidad (...
The translation of "apropos of nothing" will vary according to context, for example:
She kept smiling apropos of nothing. (Source) (Continuaba/continuó sonriendo sin motivo).
Suddenly, apropos of nothing, he said, `You're such an optimist.' (Source) (De repente, de la nada, dijo: Eres todo un optimista.)
Apropos of nothing, what do you think ...