The above phrase is commonly known as the "bismillah", because in the Arabic it reads "bismillah-ir-rahman-ir-rahim". It prefaces every surah of the Quran except for the ninth. The phrase is also pronounced at the beginning of an activity to dedicate it to Allah and receive his blessing.
However, the pronouncement of the name of Allah as Gracious and Merciful did not begin with the advent of the Quran. The previous scriptures such as the Taurat, Zabur, and Injil speak often in the name of Allah as the Gracious and Merciful or Compassionate. One example of each is given:
The Taurat -- And he (Allah) passed in front of Musa, proclaiming, "Allah, Allah, the merciful and gracious God, slow to anger, and abundant in loving kindness and truth... (Exodus 34:6)
The Zabur -- But you, Lord, are a merciful and gracious God, slow to anger, and abundant in loving kindness and truth. (Psalm 86:15)
The Injil -- Let us therefore draw near with boldness to the throne of grace, that we may receive mercy, and may find grace for help in time of need. (Hebrews 4:16)
Like the above-mentioned verse says, may you "find grace" and "receive mercy" from Allah, the Gracious, the Merciful.
Indeed all the nations may walk in the name of their gods;
But we will walk in the name of Allah our God forever and ever. (Micah 4:5)