Is it obligatory to fast in fulfillment of a vow when the time of the vow and the date of obtaining the job have been forgotten? And are there Quranic verses for bringing sustenance and relieving distress?

Your brother is not obligated to fast if he specified the vow to a certain time and does not know whether what the vow was conditioned upon occurred during that period, because the default is the acquittal of responsibility, and he is not obligated to fast with doubt. Similarly, if he conditioned his vow on obtaining a job and did not obtain it, he is not obligated to fast. It should be known that this type of vow is disliked, as it brings no benefit, repels no harm, and averts no divine decree, for the Prophet, peace and blessings be upon him, said: "Indeed, it does not avert anything, and it is only extracted thereby from the miserly," and "Do not make vows; for a vow does not avail anything against destiny, and it is only extracted thereby from the miserly."