Is it permissible to report a colleague to the teacher secretly after he threw a paper containing the words of Allah into the wastebasket?

The basic principle is the permissibility of concealing a Muslim who has committed a sin, as long as he is not openly defiant with it, based on the Prophet's (peace be upon him) saying: "Whoever conceals a Muslim, Allah will conceal him on the Day of Resurrection." Therefore, it would have been better to correct your colleague's mistake by removing that which contains the word of Allah from the defiled place, advising him gently, and explaining the impermissibility of what he did. As for spreading his deed, that is not correct.