Is seeing the phrase "Allah [God] brings relief" on a residential building after a sincere prayer for relief, for the first time in ten years, merely a coincidence or a message and good tidings from Allah?

There is nothing definitive regarding the writing you saw. Supplication (Du'a) has a profound effect, and Allah Almighty has promised to answer it. A Muslim may find in something a glad tiding for the answering of his supplication, just as it is not surprising for Satan to show him something to lead him into conceit. We advise not to pay attention to this writing, and to turn to Allah with sincere supplication, with trust in Allah's promise, seeking out times of answer, and not being hasty, and beginning and ending with praise of Allah and prayers upon the Prophet (peace and blessings of Allah be upon him).