Do you think a man like Petyr Baelish believes in the gods

Gods, I hope not. 

Because the sheer nihilism that it would take to believe in the gods despite having murdered every benefactor who ever aided you and took you into their household and then murdered your wife despite having vowed to protect her, to say nothing of all the poor saps you had murdered just for business (Ser Hugh, Ser Dontos, various Vale lords), or more precisely, to do those things despite believing in gods who would punish you for eternity for doing them, is quite frightening to think about.