Is it obligatory to inform the employer about a colleague's search for another job, considering that the colleague causes problems and the employer has delegated administrative matters to the questioner, and is informing considered backbiting or is concealing the information a betrayal of trust?

Spreading any news that its owner does not wish to be spread is considered an offense and an aggression, especially if it could cause harm to them. It falls under backbiting, which is a major sin, to reveal what the speaker or owner dislikes being made public. This is in accordance with the Prophet's (peace be upon him) saying: "None of you believes until he wishes for his brother what he wishes for himself." Therefore, you should not inform your employer that your colleague is looking for a job outside the company, because that is their right and does not harm the institution. Concealing this matter is not a betrayal of trust.